Groups with the same character table must have the same size and same abelianization,
Does same character table imply isomorphic abelianizations?
although they need not be isomorphic.
Do finite groups with the same complex character table always have the same composition factors?
For example $ D_4 $ and $ Q_8 $ have the same character table and have the same composition factors.
Another more interesting example is that $$ AGL(3,2)=2^3 : GL(3,2) = PerfectGroup(1344,1) $$ has the same character table as the nonsplit extension $$ 2^3 \cdot GL(3,2)=PerfectGroup(1344,2) $$ and these do indeed have the same composition factors. Notably there is a noncyclic composition factor.
Theorem 4 of
Kimmerle, Wolfgang; Sandling, Robert, Group theoretic and group ring theoretic determination of certain Sylow and Hall subgroups and the resolution of a question of R. Brauer, J. Algebra 171, No. 2, 329-346 (1995). ZBL0840.20006.
states that finite groups with the same character table have the same chief factors (and hence the same composition factors).
As one would expect, the proof seems to use the classification of finite simple groups.