What are the advantages and disadvantages to timber framing vs. platform framing?
I can think of one disadvantage to timber framing vs. platform or balloon framing, and that's the need of a crane.  Depending on your site, bringing in a crane to install the upper members might be a problem, whereas with stick-built, you're not moving as heavy of pieces, so it's not as much of an issue.  (it still might be useful when it comes to roof trusses, though, but it's not necessarily required)

An advantage is that timber-framed structures can span very large areas, so if you like an open design, great rooms, and the like, it's very easy to do.  As none of the internal walls need to be load bearing, you can more easily move things around should you need to add a bedroom or remove one down the road.

... but before you built, you might also want to consider comparing to other newer building methods, such as SIPs (structural insulated panels; built off-site, then get craned in and installed in over the course of a few days, so the building envelope isn't open for very long), or ICE (insulated concrete forms; install like blocks, rebar is placed, then you pump in concrete).  Both have great thermal characteristics, if you're in a climate that needs it.

With anything that's not platform built, you're also going to want to check with your local permits office -- anything that's not what they're used to dealing with is going to slow things down considerably.