I recently purchased an iPhone 6, and I used to use an iPhone 4. I have several apps on my old phone that I'd like to bring onto my new phone. I've backed up these apps into my iTunes, and I'd like to install them from my iTunes onto the new iPhone.
My problem is that, when I try to install the apps onto my new iPhone, it won't let me do so without deleting all of the existing data on the new phone. I've been using the new phone in the interim, so I've got a bunch of stuff on it that I'd rather not lose.
How can I install these older apps onto my newer phone without losing everything?