Yes you can run the Solo app and Tower at the same time.
I run Tower on an android phone wedged between the controller and my iPad mini, which runs the Solo app. I mostly use the Solo app for the live video feed and the smart shots. I use Tower for pre-planned "auto" missions. Even when I'm not running an auto mission, it's nice to have the live video feed on the iPad, while the tower app shows me where the drone is on a map. (You can do that in the Solo app by itself, but I like giving the live video feed my full screen).
If I'm running a mission in Tower I can interrupt it by hitting the fly button on my Solo controller. Then do stuff like a smart shot. When I hit "Auto" again in Tower resumes the mission from where it left off. You just have to be careful the Solo doesn't fly through a house or trees trying to get back to where it left off in the mission.
Full disclosure: I haven't done this in a couple months and there have been updates to the Tower app since that time (and the Solo app). I'm pretty confident it will still work - maybe I'll give it a test run in the next couple of weeks.
Anyway, my advice would be not to mess with Tower or the Mav Pilot software linked to above - at least not for awhile. Get used to the Solo app and it's smart shots. By the time you've got it mastered, they will probably have a version of Tower released for iOS anyway.