Difficult to say, as in colloquial language everyone has their own ideas about what certain words mean and how offensive they may be. Both words are mild insults, although of course it depends upon how they are used and in what context. Both words are only appropriate to be used towards females. Calling a man a "cow" or a "mare" will only confuse him! It would be like calling a man a "bitch". Generally speaking, "cow" is probably more offensive than "mare". Don't ask me why, that's just how it is! It may be because a cow is percieved as being a slower, uglier animal than a horse. "Cow" is probably more universal than "mare" in that you are more likely to hear it in conversation. There are regional variations in how popular one word is over another which are far too complex to get into here! That's the nature of Britain, I'm afraid. Go 50 miles in any direction and you will hear people talking in a completely different accent and using different dialect words. Generally both words would be used with adjectives, as in "you stupid cow" and "she's a dozy mare." I hope this is useful. Ask a question about British slang and you won't hear any short answers!

Both of these terms are applied to women, and generally speaking, although they are in the same vein as one another, "Cow" usually refers to an obnoxious woman, and "Mare" refers to a silly woman, although "mare" is more of a teasing name between family members, friends, etc. rather than an actual insult.

Extending Mister c's answer, in Glasgow, I have never heard someone called a mare. A cow is a girl who will have sex with almost anyone (tart, slut etc.). I have been known to call a guy a cow, although I am the only person I have ever heard use the word cow as an adjective for a guy, as it's uniquely a girl thing. I did it as he would, and did, have sex with anyone who was willing. I said it as a joke, and it got a laugh, so it had the desired effect. So there is the Glasgow definition, any more definitions anyone?

Cow is definitely the stronger of the two insults. You will find it pronounced in soaps as "Kaa", usually hurled by one drop-out female to another after it is discovered the second has been having sex with the first's boyfriend and just before there is a cat-fight. The implications of the insult are that the person is a)ugly, and b)uncaring of others feelings, even going so far as to deliberately cause hurt. A mare, on the other hand, with its connotations of a large-backsided plodding horse of little intelligence, tends to be given to someone who has done something stupid for the second or third time. A girl who continually pushes at a door marked "PULL" will earn the name from the boy who eventually shows her the error of her ways. He might also, of course, call her a silly moo (moo being the sound that a cow makes). It does appear to be more endearing.