What are the differences between Bitcoin and other crypto-currencies? What is it that makes Bitcoin unique amongst crypto-currencies?  Is it just the most popular one or does it have better design aspects too?
Until a careful analysis is performed, I think most crypto currencies are very similar in all properties except one - adoption.

Bitcoin has the largest hash rate, participating nodes, accepting sites, and people going over the code and protocol.

While alternative currencies can gain more hash rate, especially if effective forms of merged mining are developed and adopted, it remains to be seen if they gain the same adoption (sites accepting them), and developer base (strong developer community).

(See this post about merged Namecoin/Bitcoin mining)

Edit - One notable exception to this is Namecoin, which does offer unique features making it suitable to use as a DNS or general name registrar.