Finnish and Hungarian belong both to the Uralic family but they are in very distant subgroups. Finnish is Finnic, and Hungarian is Ugric. They had a common proto-language approximately 5000 years ago so we can very well assume that they are not very similar.
What always happens during the history of languages is sound changes. Finnish is a conservative Uralic language so it has retained many words in less-changed forms than Hungarian. For example, the word for son and boy was *pojka in Proto-Uralic. It is poika in Finnish and fiú in Hungarian. Sometimes, the common word has diverged into completely different forms. The Proto-Finno-Ugric word *siŋere ('mouse') is hiiri in Finnish and egér in Hungarian. Mere sound changes strongly suggest that these two languages are far apart.
Phonology is different. Both languages have contrastive phoneme length, but Hungarian vowels also have different quality depending of the length. Hungarian has more consonant phonemes, the difference is big especially in sibilants, of which Finnish has just /s/, whereas Hungarian has several. Both languages have vowel harmony, which causes that there are different suffixes for the same grammatical element, which are used according to the vowels of the stem.
They are different lexically. Those Uralic words (as well as old loans) they have in common sound very different, as presented above. Both languages have a lot of Germanic loanwords, which Hungarian has got from German, and Finnish from Swedish and its predecessors. There are not much common Uralic roots, but both languages use derivation very largely, which means that it is always possible to create new words out of the old roots.
The core of the grammar is similar. Both languages use case inflection but the number of cases is different. Finnish has more different types of infinitives. Words can be suffixed heavily with inflectional and derivational suffixes. Both languages use postpositions.
It must be noted that the historical distance between Finnish and Hungarian is big. Romance languages as well as Germanic languages began diverging from their respective proto-languages 2000 years ago. In the case of Finnish and Hungarian, we need to add at least another 2000 years to that. This fact alone tells that they are very different. Old linguistis even claimed that they are not related. Hungarian historical linguist János Sajnovic wanted to prove that they belong to different families. So different they are. However, when he had collected evidence, he proved that they really were born from the same proto-language.