To say Star Trek is inconsistent with time travel would be a massive understatement. Each story involving it usually has its own rules which are established or inferred during that story. This is part of the nature of having a science-fiction dynasty where dozens of writers contributed to the canon.
That said, the writers of the film Star Trek sought to reboot the franchise while respecting the original canon we all enjoyed. As part of the reboot, they had Spock and Nero's actions create a new timeline that's similar, but crucially different from the original. This was meant to be irreversible, as to establish a new canon. In addition, they sought to do this in a manner that shows the original canon still exists.
The mechanism they used for this was Spock. By having the original Spock get hurled back in time into this alternate timeline, the writers show us that the stories we grew up on are still there, but they're inaccessible to this new Enterprise crew. Otherwise old Spock wouldn't be the Spock we knew from TOS and TNG, as his memories would have been presumably overwritten with the new historical events of the film. During the film, there's a scene on the bridge where they establish this in-universe:
Spock: Nero's very presence has altered the flow of history, beginning with the attack on the U.S.S. Kelvin, culminating in the events of today, thereby creating an entire new chain of incidents that cannot be anticipated by either party.
Uhura: An alternate reality.
Spock: Precisely.
In addition, the makers of the game Star Trek Online, which takes place in the original timeline, released this image which helps clarify where their game fits into the new status quo.
The 'Alternate Timeline' being the one of Star Trek (2009). This nicely lays out how the two timelines are related, and shows they both continue to exist.
