On my work iPhone 4 I have apps like dropbox, logmein, office stuff
You can have the same content on each device.
The reason why you "may" have to first delete the existing backup, is that iTunes keeps separate backups, by device. Even if you "wipe" one of the phones first(you'll have to do that to restore from the backup from the other phone), unless you delete that phone's backup in iTunes, iTunes may not give you the option to restore from the other phone's backup, you "may" only be presented with the option of restoring from that phone's existing backup. I say may, because I've never run across a situation like this and experimented with it. Quite frankly, it seems like a lot of work for very little gain & the real risk of actually losing data. I think you'd be better served by putting the same apps on each phone. You could do that even if you used different iTunes accounts at both work & home-easier if they're the same account.