Looper indeed has its unique way of looking at the concept of time travel. Many people have praised the way the concept is depicted here, yet they have also indicated that this approach allows for quite a few plot holes.
The most obvious ones arise from the fact that the story seems unaffected by so-called time paradoxes. A prime example of this is the final twist: young Joe realizes that old Joe is about to kill Cid's mother Sara, which will be the event that causes Cid to become the feared Rainmaker. So young Joe sacrifices himself, and by killing himself, old Joe also vanishes. Sara lives, Cid never becomes the Rainmaker, and the dark future is prevented. But this immediately causes a paradox: if young Joe is dead, then he can't age to become old Joe. If old Joe doesn't travel back to the past, he can't set in motion most of the events in the movie that cause the finale, including young Joe killing himself. And if young Joe doesn't kill himself, old Joe will be sent back in time, forcing the world to go into an endlessly repeating cycle of events.
This is often referred to as the "Grandfather paradox", named after the thought experiment where one cannot theoretically go back in time to kill one's grandfather: doing so would prevent you from being born. If you're not born, you can't travel back in time, and thus you can't kill your own grandfather, which, in turn, causes you to be born, travel back and kill your grandfather. In short, a paradox will ensue when one travels back in time, and does something to affect the act of the time travel itself.
In an interview, Rian Johnson, director of Looper, has indicated that it was part of an early creative decision to take a linear approach to time, instead of one where there is a constant switch to alternate timelines. This means that time is dealt with on a moment-to-moment basis ("nothing has happened until it happens"). In practice, this means that any changes made to the timeline do not retroactively affect the entire timeline; they only take effect from the moment that changes were made ("Everything is kind of being created and fused in terms of the timeline in the present moment; the universe would realize [the change and] catch up with it").
For example, as soon as young Joe starts to carve messages for old Joe into his arm, Joe receives the scars from this change. Technically, he should have had them even before he was sent back in time. So the timeline is not completely re-written; the change has no effect on the past (before the change), only the future (after the change).
In Johnson's own words: "What's important is what the experience of these people are in the events that happened in the movie. And that, you have to experience in a linear fashion." Whether or not this is an accurate depiction of time-travel, it does conveniently negate the effects of time paradoxes.