The Russian mob has a hand in everything important in Russia. If you go as a tourist, you'll probably feel fairly safe and nobody will notice you. If you go as an investor and try to start a company you will most definitely deal with the mob. It will probably start with you dealing with the police and other 'officials' who are actually a front for the mob.
Isn't it more likely that tourism is also a big business? Tourists only come back (or tell others to go) if they feel safe. Asking tourists for protection money or other such things is just shooting yourself in the foot if you (indirectly) get money from tourism.
Right. In other words if you go there as a tourist you will most likely get no problems from the mob. They might own the hotel, but you'll be mostly safe. You won't have to bribe them, you won't have to have a bodyguard with you.
If you go there ready to start a business or invest a couple million dollars, you'll definetly deal with mobsters.