What are the fundamental differences to the total count of coins that can be mined or the process of mining?
Obviously BTC was first but are the subsequent alternatives more refined or more efficient?
Nope. Some coins have technical innovations (these are basicly the only interesting altcoins.)
These altcoins may be more efficient/refined but not by design (but because the blockchain is small etc.) There's still nothing _better_ than Bitcoin. And if there will be, Bitcoin can implement it.
I am personally waiting for Nimblecoin (DECOR+GHOST protocol implemented, giving very high transactions per second rate of 2000+)