Different string brands can have different tension even when the gauge is the same.
Yes, higher string tension means that the neck is under more strain. If you go to TI flats, your neck will be under less strain due to the lower tension. It is possible that your truss rod will have to be adjusted. Note: do not attempt this yourself if you don't have any experience. At the very least, do some research to find out the proper way to adjust a truss rod: adjusting a truss rod the right way is easy, but adjusting it the wrong way is also easy.

Doing it wrong could damage your neck.