There is a distinction. Usually statistical mechanics means the ensemble theory and partition functions that connects microscopic systems to macroscopic ones from material point of views. However, statistical physics is a bit more generic, for example complex networks may not use ensemble theory or partition functions and could use only statistics on the network, such as average neighbourhood or similar.

Kinetic theory stuff for one, like deposition, growth, sandpile type things. Complex networks and lots of dynamics stuff falls under statistical physics umbrella but not statistical mechanics. Stat mech's amazingly wide applicability makes it easy to think it's THE approach to approaching things statistically, but it's not. The broad encompassing approach has a name, statistical physics.