I know that it depends on what you're trying to achieve, but in general we use fresh thyme as much as possible. It has a lovely aniseed flavour and also helps to improve the flavour of many dishes. However, if you need a little more punch than that, we use dried thyme.
I've grown both types and prefer the variegated. The green leaves are more tender and have a sweeter flavor than the purple-leaved variety. I also use the Greek thyme, and I think it's better than the French variety, because it has a stronger flavor.
The flavor of thyme is very dependent on the variety of plant you are using. Most often, it is used for its fresh, clean, and somewhat delicate flavor.
Some people will maintain that there is no difference between thyme varieties, they are all the same species. They are all easy to grow and they all have similar uses.