Sign in with the connection currently using WiFi instead of GPRS/3G, and it will keep the WiFi hardware on, which is what kills it. There is no way for a normal user to know that they need to open new, persistent connections over a connection type that uses the cellular modem instead of the WiFi. And Android defaults to using WiFi for new connections if the WiFi hardware is powered on and connected to a network.

The best way to save battery on an Android phone is to turn off WiFi unless you need it at some point during the day. iPhone will stop attempting to do push notifications and the background Apple apps will disconnect if the only available connection is WiFi.

I'd also like to point out that "I bet the app was poorly written" is exactly the situation Apple is currently avoiding until they are confident they have a reasonable technical solution to it.