Both the set and the group operation are different.
The first group includes things like f(x)=sin(x)+2, which is not invertible.
While the second includes f(x)=x, which is not in the set of the first group.
Also the group operation for the group of invertible real mappings is (usualy
though you have not said so) function composition, that is (f o g)(x)=f(g(x)).