The primary advantage of common law is that statutory law often takes a long time to change and adapt, and it's nearly impossible for the legislature to take into account every possible consideration in the present and the future in regard to defining the law and how it will apply to people who actually interact with it.

The secondary advantage of common law is that it allows people to ask for reasonable compensation when they encounter a new situation, without having to wait for the legislature to identify, debate, and pass a new law with regard to their specific situation -- which may be so specific that no law would ever make it to the legislature for consideration.

The third advantage of common law, at least in this day and age is that there is almost a thousand years of case law that can be read, analyzed, and referenced when making a determination of what the "right" outcome of a case should be (English common law is generally accepted to have begun around 1189 AD).

Now, for the disadvantages. The primary disadvantage of common law is that the laws aren't all stated clearly and plainly in unambiguous and structured language, like statutes are in civil law countries. This means that if you're seeking information on a common law cause of action, or you're a party to a matter in which the common law is coming into play, you almost absolutely need the advice and counsel of a licensed attorney, who knows where to find the law, how to read the law, how to know what's good law (or what's been overturned), and how to argue for whichever interpretation of the law you want to apply in your case.

The second disadvantage of common law is that it is made by judges, whom many people argue should have the sole purpose of enforcing the law, not creating it. Any common law jurisdiction opens the door for judges to leverage their equitable and appellate powers to alter the rules, either explicitly or implicitly, and to add or subtract from the statutes that have been passed based on their inherent power to "create" law under common law principles.

There are probably more disadvantages, but those are the two big (and most common) issues that people find with such systems.

Disclaimer:

This answer is not a substitute for professional legal advice. This answer does not create an attorney-client relationship, nor is it a solicitation to offer legal advice. If you ignore this warning and convey confidential information in a private message or comment, there is no duty to keep that information confidential or forego representation adverse to your interests. Seek the advice of a licensed attorney in the appropriate jurisdiction before taking any action that may affect your rights. If you believe you have a claim against someone, consult an attorney immediately, otherwise there is a risk that the time allotted to bring your claim may expire. Quora users who provide responses to legal questions are intended third party beneficiaries with certain rights under Quora's Terms of Service (http://www.quora.com/about/tos).