This is a great question. I'll try to expand this answer later on, because I have bunch of examples that come to mind.

For one thing, as with many synonyms in English, one is of Germanic origin ("split") and the other is Latinate ("divide"). This creates the ground for the inclination that Robertd has to hear "divide" as slightly more formal. I would probably say something like "split" carries with it coarser connotations of instances of division (as in splitting a log), and "divide" carries with it a more granular connotation of instances of splitting (as in mathematics).

Note that I am using granularity as the measure for comparison and NOT refinement because I think the attempt to assign "high and low" speech alternatives to this case (and others like it) is erroneous. The poetry of Robert Frost, for example, and many others is actually heavily invested in reclaiming the "high" use of the "older," "English" words and in using them insistently and purposefully.

As with many of Germanic/Latinate word pairs, because of English speaking people's investment in assigning and maintaining shades of meaning to each option (to the extent possible -- real and imaginary), there are actually quite a number of cases in which the one would be very awkward to substitute for the other -- at least relative to the standards of a 'native' speaker.

We might do well to make a community repository of them. =) ...I'll add some more to the examples already here later on.

Used as verbs they are generally synonyms, it may depend on context. Used as nouns they can also be synonyms, but it is not difficult to find differences.

For instance:

As a noun split can mean

A bottle of wine containing 0.375 liters, 1/2 the volume of a standard .75 liter bottle

A divide can not mean A bottle of wine containing 0.375 liters, 1/2 the volume of a standard .75 liter bottle. But you can say: There is a great divide between us.