Difference between subshell and sub-subshell?
As you pointed out the the s, p, d, and f denoted subshells of atoms. 

The term sub-subshells is very uncommon. I did find a couple of uses. 

Materials Science
By G. K. Narula, K. S. Narula, V. K. Gupta

pdf file about Modern Atomic Mechanical Theory, page 24

What the authors are noting is that the 2p subshell will have three orbitals which can support two electrons each. Each of these three orbitals is being referred to as a sub-subshell.  

As I said it seems a very uncommon term, and I'd suggest you avoid using it.