Just a couple of points here:
1. The time is relative, so in reference to the comment that the two aircraft going opposite ways around the Earth would differ because one is moving "with" the Earth's rotation, that would be fine if you were comparing it to a clock at some "observation point" outside of the system.
2. The point that there is less gravity at altitude having more affect is quite valid, although in the one example we were comparing two aircraft so ostensibly that fact is washed out.
3. The differences from a relativistic viewpoint are virtually impossible to measure, but if we were able to compare the two there might be a difference as the westbound aircraft is going to be travelling a lot slower across the ground due to the the winds, so compared to a ground based clock, one has moved faster than the other.
Of course, problem with that is that both aircraft are moving with the wind, so relative to _each other_ the effect of wind should wash out. I'll leave that to you to ponder, as I need to shower now and have no time to write anymore!