The culprit is PowerExpand, not Simplify. The two forms s1 and s2 are not totally equivalent. Also, you should expect trouble when inverting a function that is not invertible over the complexes:
Solve[s1[a, b, tt] == x, tt]
You will see that there are two branches. It seems that InverseFunction chooses the first one, which involves the logarithm of a negative number (if a>0). With s2 the formulas are different:
Solve[s2[a, b, tt] == x, tt]
and InverseFunction makes its choice.
The output of InverseFunction should be carefully sanity-checked each time.
