Full Adder vs. Half Adder
The C's are not the same, but your statement about the AND gate is right.

Unfortunately, the people who drew your full-adder decided to save space instead of maximizing readability. This image does a better job of showing the relationship between half adders and full adders, even though the circuit is identical to the one above:

 

If they'd just put the wire splitting points inside the grey boxes showing the half-adders instead of outside, they'd be perfect.

Here's a block diagram you can use to see the exact relationship between a half-adder and an adder.  It shows clearly something I find quite interesting about adders.  The 'or' at the end looks like it could swallow a carry if both half adders were to emit a carry bit at the same time.  It seems like we might need another half adder to resolve this, in a never ending chain.  However, if you run the truth tables for the half-adder, you find that it is impossible for both half-adders to carry at the same time.  Rejoice!