Except it doesn't really. You still have to deal with the important things. All that's removed is the tediousness of the physical server maintenance. You still need to architect for redundancy, deal with latency, client/server caching, etc. Your app cares just as much about the server as it always did and your server-side code needs to be as robust as if it were running on dedicated hardware (perhaps more robust).

And as wmf pointed out, we already have plenty of buzzwords for this. PaaS, IaaS, "Cloud computing", etc.