This one came up in a CAE class. As far as I can work out there is in fact no difference at all in the literal definitions of stumble and trip. They strike me as being synonymous, which you can clearly see at the dictionary.reference.com:
It is certainly not true that you need a stimulus to cause you to trip – I’m sure that all of us have at some time or another tripped for no good reason at all. No obstacle, no stimulus, just our own clumsiness. Of course, you could say you tripped over your own feet, but certainly no external stimulus is required.Nor can it be said that a trip usually leads to a fall, simply that it may lead to a fall. As indeed may a stumble.
However, we would use “to stumble” to describe how a drunk or very tired person might be moving and that there is no sense of trip in that usage.
And of course, yes, a horse can also stumble 🙂