In my opinion, the main impact this would have is that presidents would not be able to maintain their influence so far into the future as they can now. A second impact would be that more presidents would be likely to get chances to nominate a justice or justices.
As things are now, a president can nominate a justice when he or she is quite young. For example, Pres. Bush nominated John Roberts when Roberts was only 50. Now, Roberts is likely to be on the Court for at least 30 years and Bush's influence will still be felt 25 years from now. Pres. Obama has done the same thing with Elena Kagan -- she was 50 when nominated.
If justices had term limits, each president would be more likely to get a chance at nominating a justice. A justice could not stay on the Court and only retire when a president they liked was in office, for example. This would be fairer, I would think, because it would take the element of random chance out of it -- why does one president deserve to appoint justices more than another one?