When I "Leave Animus" I am able to go outside and explore the modern day "Villa Auditore" as Desmond. There was an early mission to do some stuff out there, but otherwise I don't see much point and apparently if I stay outside too long something bad happens, though I don't know what - I've never waited.
I doubt this is a spoiler, but just in case:
What I have noticed is that there is a big stream of footprints that I can see with eagle vision. They lead from the stairs near the fountain directly to the spot where we have the animus set up. I've checked them out about a half dozen times, but I can't find any significance. Do they become important later - or do they basically have no importance?