Hi all.
well Kevin it is not exactly like that.
Normally you build counters using flip flops. According to how you will connect the clocks of your flip flops, you can get a sync. or an async. counter. You have a syncronous counter when all clock inputs of the flip flops are connected to a single, common clock. An asynchronous counter uses each flip flop's clock differently. You may want to take a look at this page for more details and some examples:
http://www2.ele.ufes.br/~ailson/digital2/cld/chapter7/chapter07.doc5.html
As for advantages and disadvantages, each counter fits in different applications and therefore I don't think you could make a comparison between them. In general you put sync. counters in synchronous systems and asynchronous in async. systems (these are the faster ones btw., but are more complex to design)