OP, the difference is the nature of the plots. On a basic level, in oldTrek the whole audience perspective was usually given from somebody travelling back and involving themselves in 'fixing' the splinter reality so that it re-aligns with something they already know. I guess NuTrek takes the tack that nobody fixed this one, so it continues on parallel to the original instead of merging back into it.
Admittedly oldTrek (usually) insinuated that there is only one timeline and the reason repairs 'needed' to be made was that if they didn't then the original ceased to exist. But while this was the prevailing view, exceptions did occur. The "fight your fate!" splinter of TNG: Time Squared
, or the whole underlying premise of TNG: Parallels
which basically contradicted the normal way these things were presented to us in oldTrek. TOS: Mirror, Mirror
and all of its DS9 and ENT sequels would also seem to confirm the idea that 'parallel', rather than 'replacement', timelines exist within the Trek causality.
So in summary: Trek time travel logic is basically whatever the writer decides it is at the time.