I've now finished reading Madame Bovary. I enjoyed it reasonably well. I now can't remember the tone of Anna Karenina as well, so comparing the actual writing in the two books is difficult. There are obvious plot parallels between the books, at least if we only look at the stories of the title characters (AK has more different plot lines).
An obvious plot difference is that the people in AK are of a much higher social stratum. Both books deal with money issues, but very differently. People like the Bovaries are basically invisible in AK, which deals with aristocracy (close up) and peasants (as accessories).
But I guess the biggest difference, for me, is sort of like the difference between where I work now and my old job - the people in Anna Karenina are smart! Anna herself is actually an intelligent, strong woman. (She reminded me of Sally in some parts.) Emma Bovary is...well, not necessarily completely stupid, but definitely weak.
When you read Madame Bovary, it's a bit like watching a horror movie - you want to yell at the characters about the stupid things they are doing. And, of course, you are forced to listen to a lot of rather idiotic dialogue. (I could have done without Homais the Chemist entirely.)
The characters in Anna Karenina are mostly smart and strong, so the things they do fall more easily along a kind of intellectual or moral axis. You watch them make choices that are more moral in nature. And the non-plot conversations in the book are far, far more interesting. Tolstoy seems to actually respect most of the characters, where Flaubert was obviously making fun of many of his.
I wouldn't really tell someone to go read Madame Bovary for fun, though I didn't hate it or anything. I still recommend Anna Karenina to all but the most determined haters of the (broadly-defined) genre.