Because you're given a brand new IP address from a brand new router (in this case your phone) and you're on a totally different network. It's not something they *could* build into the Wifi protocol, it's a routing issue on top of a TCP/IP issue and there's simply no device-side solution to the problem you described. Its up to the server to manage the session for that connection, Rapidshare would need to design their infrastructure and connectivity to support interrupted sessions and that sort of roaming, essentially giving you some sort of cookie that expires after say, 10 minutes, and regardless of your IP address/network would let you resume your download.
They specifically dont do that because their business model is built around promoting premium accounts with advanced download features (like resuming interrupted downloads) and unlimited downloading. A session cookie system like this would be rampantly abused by people with premium cookies sharing them with people so they dont have to pay for premium accounts. In fact, cookie sharing already goes on quite frequently with people just looking to get free premium access even

If you're downloading via bittorrent, for example, and you do that handoff you still lose connectivity for a moment, but the bittorrent client still has that list of IP addresses you were connected to and just reconnects to them because the protocol, the client, and the server (the tracker) were designed with this functionality in mind.
The only way to do a truly seamless WiFi handoff is if your handing it off between two access points on the *same* network, such as in an office building, and only then with higher end access points that are properly configured. We've got a system like this at work, and I can carry my laptop around the whole building and not skip a beat even with a download going. It just hops right to the next AP with the stronger signal when they overlap.