CAT 5 and CAT 6

You don't need different connections. They both use RJ-45 jacks. You just need to make sure that whatever you are plugging into supports Gigabit Ethernet. Though I would gather that you'd have to be running on a pretty old box not to support it.

Are you running Cat 5 or 5e? 5e is the same throughput as Cat 6 - Cat 6 is considered better performance. You can also find Cat 7 but that's not a recognized standard.
 
Like Calculagator said, unless you have a need for CAT6, there's no real reason to replace. When I purchase new cables, I purchase CAT6, but I don't feel a need to replace CAT5e, which is good for 1gb at 100m, but CAT5 is only good for about 30' at 1gb.