The main differences between terrorist groups are going to be who/what they target and why. There will also be some differences in how the targets are attacked. Some terrorist groups do not wish to harm anyone. They simply want to scare people into paying attention to their cause. Other terrorist groups want to kill as many of their perceived enemies as possible. The goals of a terrorist group as well as their particular beliefs are going to strongly effect the target and type of their attacks.
Terrorist groups have different methods than they did in the past. The internet and social networking have helped them get their message out to a large group of like-minded people in ways they never could before. This also makes it much harder to hunt terrorist groups, because the finance stream can come from sources all around the world.
We should also be aware that the Left-Right model is different in Europe than in the United States. A Left-Wing terrorist group in Europe might have very different goals and ideology than one in the U.S., and could easily be misidentified. As usual, Wikipedia has a very comprehensive listing of the verious flavors of terrorism; check the sidebar on the right for many articles on different ideologies.
Left-wing terror groups do not get as much attention anymore as they used to, but they still exist in many countries. In years past, the Red Army Faction in Germany and Action Directe in France were much feared, but not anywhere near as much as Shining Path in Chile, which killed thousands in terror attacks. Some terror groups have left or right wing affiliations along with broader ideological goals. One example of this might be FARC in Colombia.
Right-wing terror attacks in recent years have not been carried out by groups (Timothy McVeigh was loosely aligned with a right-wing militia) but more by individuals, e.g. the lone attacker in the Norway attacks last year.
The major difference is in who the terrorists attacks would be aimed at. Ideological and religous terror would be aimed at people who are enemies not because of "tangible" factors but because of philosophical ones. This is why, for example, Al Qaeda has put more effort into attacking the West than into attacking the governments in its own area.