It's basically the Back to the Future II thing. Or the Terminator thing...
The moment old Janeway went back in time, the world she came from changed and ceased to exist, as we know she was successful. She continues to exist because she's the time traveller, shielded from changes.
That's how the internal logic of these stories goes anyway.
WOW.
Okay, first of all, Back to the Future and Terminator have vastly different takes on time travel, and both of them are different from Star Trek.
In the Terminator, it seems that time travel is effectively meaningless: people with knowledge of the future try to change events but are ultimately thwarted, often bringing about the very events they were trying to avoid. In the first movie, we have a closed loop: John Connor sends a man back in time to protect his mother, and that man is his father, indicating that all that happened in the movie is what John Connor remembers as his past. We even have the photo that Kyle Reese has of John Conner's mother, which is the same photo taken by the boy at the end of the film.
In the second film, we learn that Sarah Conner tried to warn people about Skynet, but not only was she Cassandra-like in her prophecies (that is, doomed to not be believed), but it was research on the wreckage of the remains of the first terminator that gave Cyberdyne its technological edge. And so on.
Back to the Future has in common with Star Trek the notion that a time traveler who changes history, upon returning to his own time will still remember events as they were, rather than the events caused by his changes. But it differs from Star Trek in changes to time traveled artifacts (including the time traveler himself) will occur slowly, or occur based upon probability. Thus, as Marty McFly makes his parents meeting less likely, first photos of him and his siblings, and later his body itself, begin to fade. And the success of efforts to save the life of Emmett Brown in 1886 can be judged by looking at an 1886 newspaper article about his death brought back from 1986. Star Trek, however, seems to operate on the notion that the newspaper article that was actually brought back in time will not change, but the copy of that article that exists in the library in the future will change, and that change will be sudden and complete.
(Tiny side bit: I did a little chart of the parallel universes at play in the Back To The Future movies, and it left me with this lingering question: what happened to the Marty McFly who grew up with the cool dad who wrote science fiction? As far as I can tell, he travels back in time in the Delorean and is never heard from again?!?!)