Personally, if I were to have only one, it would be the stoptail. The PRS tremelo is very stable (I upgraded with FU tone springs) though it is still a tremelo and thus I have to make minor tuning adjustments more often. As you already have several trem-equiped guitars, this is not likely a surprise to you.
I expected the stoptail tone to be different from the tremelo simply from the fact it does not have a big hole cut in the back of it. If there is a tone difference I do not hear it with my playing (high gain: punk/metal/hardcore). The stoptail feel a little heavier though that is to be expected with more wood and this will vary between even the same model. Moving between the two is seamless though the stoptail's action feels slightly higher than the tremelo, not so much on the fretboard, but it feels higher off the body.
As I expected, everything feels very similar between the two models. I keep the stoptail in E-standard and the tremelo in D-standard, which gives me more reason to play both
In closing, the Core Tremonti is a fantastic model and both versions are a real joy to play. The body, neck, and pickup configuration make this an amazing playing guitar.
If you already have other trem-equiped guitars and are comfortable with that system, then you cannot go wrong with either version. I never use the tremelo and strongly prefer fixed bridge guitars which lead to my search for a stoptail. It feels more natural to me and I can discern little tone difference.