First, make sure that you are getting a true Fiber connection in the case of Airtel and not a VDSL connection in name of Fiber. Because Airtel names both of them as Xstream Fiber. I had a VDSL connection for 3 years and initially, it was called V-Fiber, and later it was renamed as Xstream Fiber. Jio in any case will only provide a true FTTH connection.

There is no comparison between VDSL and Fiber.

Between true fiber connections of both of these companies, I am giving a pointwise comparison as per the following

Jio Fiber Plan glimpse

Airtel Xstream Fiber Plan glimpse

1. Plans: Most of their plans are comparable now. Airtel has renamed its plans recently. Airtel starts from 499 (Basic) for the 40 Mbps plan against Jio’s 399 (Bronze) for 30 Mbps. The next one for 100 Mbps speed Airtel charges 799 (Standard) against Jio’s 699 (Silver). The next plan for both is 999, here Airtel provides 200 Mbps (Entertainment) while Jio 150 Mbps (Gold). Then next both have 1498 vs 1499 for 300 Mbps (Professional vs Diamond). Jio has a 500 Mbps plan for 2499 (Diamond+). Airtel doesn’t have a 500 Mbps plan. Both of them offer 1 Gbps for 3999 (Infinity vs Platinum). Then Jio has one extra plan of 1 Gbps for 8499 (Titanium). Airtel has recently added 3, +TV plans, in which they offer bundled DTH services. Their details are in the screenshot above. which in my opinion is similar to their offering of Airtel Black Plans. I am not going into the details of either since they include details of TV and Postpaid cellphone service. Which is beyond the scope of this answer. If anyone wants its details or analysis, I may add that in a future edit. GST at 18% is extra to all the plans. You can opt for long-term advance rental for any of these plans. 6 months advance rental would give you a 7.5% discount while a 1-year advance rental would give a 15% discount. I in fact am using Rs. 999 plan on a 1-year advance rental. Airtel plans are all postpaid. Jio prepaid plans are for 30 days cycle. Earlier Jio only offered Prepaid plans. But nowadays they are also offering Postpaid plans with almost similar structure. However, they are not truly postpaid per se, because you have to pay an advance rental (3/6/12 months), unlike the Airtel where you pay the bill at the end of the month. Jio Fiber postpaid basically means you will get Bill at the end of the month, which in the case of Prepaid you only received an Invoice of Payment.
2. Data: All these plans have so-called unlimited data barring Jio’s top Titanium Plan for 8499 which comes with 6600 GB of data. Here Unlimited means 3300 GB for Jio Fiber and 3333 GB for Airtel Xstream Fiber. I think this data is sufficient for the vast majority of users until 8K streaming becomes a thing.
3. Installation Charges: Right now Jio is giving a 1-month free trial offer. Where you have two options; either you pay Rs. 2500 for connection and get ONT Device and Set-top box with a 150 Mbps Speed plan and 14 Free OTT Apps. The second option is, where you can just choose to pay 1500 and get only the ONT device and no Set-top-box. I would advise for this, only if you want to use lower-end plans of Rs. 399 and 699 in foreseeable future. These plans don’t have any OTT Apps subscription. If you chose to cancel your subscription, do it before the month ends. They will refund your entire amount. In Airtel, if you opt for a plan higher than the base plan of Rs. 499, and if you pay 3 months advance rental, you will get free installation and free ONT-Router. You can pay 1500 ‘Refundable Security Deposit’ to get the Xstream Set-top box. But I would advise highly against this. This device comes with lots of ifs and buts. You would be better served by getting a third-party Android TV box or Stick like Amazon Fire Stick or Google Chromecast if you wanna consume OTT content. I have heard, in smaller cities, they are charging Rs. 1000 and extra length wire charges which is not the case in my area in Noida. You have to check with Airtel for that.
4. Installation Time and Ease: If you have Airtel Fiber Termination Box in your vicinity, they don’t take more than 3–4 days to connect you after you put a request through their site or app. If they don’t have it in your vicinity but in your general area, it might take them slightly longer and they might charge you for the extra length of the cable. However, in the case of Jio, there is absolutely no guarantee that someone will respond to your connection request as their backend system is not really good. I had Jio Fiber in front of my block so I applied for Fiber several times. But for the initial few times, I didn’t get any response or follow-up other than a text message at the time of applying. So in this regard, Airtel is much better. I would like to emphasize here that, please don’t believe Airtel if they say they will upgrade your connection to Fiber in the near future. Right now you take a VDSL connection. They probably won’t do it soon. I have been a victim of this tactic myself. I got my Airtel V-Fiber connection which was VDSL based in 2017. After 3 years of them giving deadlines after deadlines, they didn’t upgrade to Fiber. So finally I switched to Jio Fiber in mid of 2020.
5. Speed & Ping: Though in some plans speeds differ for them like the 999 plan gives 150 Mbps in Jio and 200 Mbps in Airtel. But where both have the same speed mentioned for a certain, Airtel will have 10 to 15% higher speeds than their plan speed in most cases for a true Fiber connection. While the reverse is true for Jio Fiber. For eg. for a 100 Mbps plan, Airtel will give download and upload speeds in the range of 105 Mbps to 115 Mbps while Jio will provide somewhere 90 to 95 Mbps. Airtel also seems to have better pings than Jio. It will matter a lot if you play online video games. This is an observation of my friends. I have played a lot of CS:GO using the Jio Fiber connection. My ping remains quite stable 99% of the time. It remains in the range of 30 to 40 ms which is quite good. Now since I installed Airtel Fiber earlier this year. I have noticed that my ping has marginally improved. As per my research and input from some friends, Jio seems to have a double NAT situation where certain games are not able to be played on it. This issue arises in Console Games sometimes. I have never faced this personally in any of the multiplayer online games I play.
6. OTT Content: Jio wins hands down. There is no comparison at all. Jio will provide you with a complimentary 16 OTT Apps subscription in their Gold plan of Rs. 999. This includes Prime Video, Disney+ Hotstar, Sony Liv, ZEE5, Voot Select, Voot Kids, Sun NXT, Hoichoi, Discovery+, Universal+, ALTBalaji, Eros Now, Lionsgate Play, ShemarooMe, JioCinema, JioSaavn. They also have Jio TV+ in their STB where you can watch lots of Live TV channels if you have a separate DTH connection from service providers like Hathway. Their next plan Diamond 1499 includes Netflix Basic in addition to these apps. Earlier OTTs were only available with Rs. 999 plan or higher. Now Jio has started giving OTT bundles of Rs. 100 (6 OTT: Universal+ ALTBalaji, Eros Now, Lionsgate Play, JioCinema, ShemarooMe, JioSaavn) and Rs. 200 (14 OTT: Disney+ Hotstar, Sony Liv, ZEE5, Voot Select, Voot Kids, Sun NXT, Hoichoi, Discovery+, Universal+, ALTBalaji, Eros Now, Lionsgate Play, ShemarooMe, JioCinema, JioSaavn). JioSaan and JioCinema are counted as 1 OTT. Airtel has a very limited subscription of OTT apps. Airtel provides you with a subscription to Wynk Music, Shaw Academy, and Airtel Xstream app with all plans. Airtel Xstream app has inbuilt content of different apps like Voot Basic, Hungama Play, Lionsgate Play, Eros Now, Hoichoi, Shemaroo Me, and Ultra. In Airtel, Rs. 999 and higher value plans also have Disney+ Hotstar, and Prime Video subscriptions as well. Earlier Airtel didn’t have Netflix in their OTT bundle. But they added it now. Their Professional Plan 1498 and Professional+TV Plan 1599 will give you Netflix Basic. In both Airtel and Jio Fiber cases, you can pay the difference amount and upgrade to the next tier. For example, if you opt for Airtel Professional Plan you will get Netflix Basic which is valued at 199. If you pay the difference amount you can upgrade to a higher tier (Standard or Premium) of Netflix Plans.
7. Service & Reliability: Airtel has better overall reliability. This is due to the fact that Airtel is a long-term player in the market while Jio is a newcomer in this segment. They are still finding their bearings. So Airtel will have much fewer problems related to Speed and downtime. Their engineers are very quick to respond to complaints of the user. Their customer care is also very easy to reach. Their Xstream Fiber app will let you diagnose the issue yourself, and if not solved, it will assign an Engineer in a very short time. And in most cases, they attend to the complaint in 1–2 days and resolve it. The same can’t be said about Jio. Their customer care is totally unreliable. If you face any problem it might take more than a couple of days of waiting to resolve the issue altogether.

Below are the results of my speed test when I first installed it in an offer period of 1 month. At that time the offered speed was 150 Mbps, after that, I switched to Rs. 699, 100 Mbps plan.

You can see that download speeds are above 140 Mbps most of the time but upload speeds do fluctuate a lot.

Edit: I have gotten Airtel Xstream Fiber installed just last week. I have opted for their Entertainment Plan of Rs. 999 p.m. which has a plan speed of 200 Mbps. However, as I said above, I am actually getting speeds around 240 Mbps more often than not. Moreover my CS:GO pings are marginally better than my Jio Fiber connection. For testing Airtel Fiber speeds I would highly advise not to use the Ookla app or Speedtest by Ookla - The Global Broadband Speed Test or Speed Test - TestMySpeed. They will give you a false high which could be as high 900 Mbps. You would think, that Airtel is giving you 1 Gbps speed in 200 Mbps plan. Which is simply not true. Surprisingly Ookla is quite accurate in testing Jio Fiber speeds. Which I have shown above. This leads me to believe there is some understanding between Airtel and Ookla since their ads in past regarding speeds of 4G networks have mentioned Ookla ad nauseum. Hence please use Internet Speed Test or Google’s default speed test. They give you an accurate results. Since in any case download speed will be around 240 Mbps, i.e. from Steam or Torrents etc. Below is the result of the Speed test using Internet Speed Test

Here is result of Ookla Speedtest app

Edit 2: I have updated details of Airtel Plans and Jio’s new OTT bundles.

Edit 3: I upgraded to Airtel Xstream 1498 Plan with 300 Mbps advertised speed. In Ookla Speed Test App, speeds are shown correctly now. Speed are normally above 350 Mbps to go as far as 500 Mbps actual speeds in Steam downloads. See below screenshot for my speedtest history with this plan.