When you install on the new device, you will be asked if you are an existing or new user. If you choose existing and login w/ the same email and password as you did on your old phone, it should immediately start syncing content.
One of the very strong benefits of the Rex solution is that everything syncs back to their server, so unless the phone got soaked while you were doing a bunch of edits, it should have all your tasks
. I say this as a paid account, but I don't think the behaviour is any different while you are in trial mode.