In the ordinary sense, the two are synonyms, meaning they essentially mean the same thing.
However, in a more technical aspect, the homily is an informal talk given in a church about a moral or religious subject. I have been told, though I have never been able to confirm this, that the homily is supposed to specifically discuss the Gospel lesson from the day's Mass.
The sermon, on the other hand, is a much more formal speech given at the Mass. The same source that told me the homily is supposed to be about the Gospel told me that the sermon can be on any appropriate subject.
If you look up the two words in the dictionary, you generally find that a sermon often is of the nature of a moralistic rebuke, that is, a fire and brimstone speech. The homily, on the other hand, is kinder, nicer, and folksy.
I've often noticed that all of the traditional missals I've ever seen don't even mention a sermon or homily. Immediately after the Gospel is complete, they go directly to the Creed. This is because the sermon is not really part of the Mass. However, in the instructions on how to use the missal or explanations of the Mass found in the beginning of the hand missals I have, they do mention that the priest gives a sermon after the gospel.
However, in all the hand missals and "missalettes" I've seen for the Novus Ordo (and even for the 1965 hybrid Mass) after the Gospel they tell us that the homily follows.
I presume, though I do not know, that the Conciliar church went with the word, homily, in English-speaking areas because it doesn't sound quite as harsh or restrictive as the word, sermon, plus the fact that priests had pretty much stopped preparing anything of real substance and only wanted to talk about luv and stuff of that nature. Jello is more comfortable with a homily than with a sermon.