Sticky situation is an informal way of saying that the situation is embarrassing and makes you feel worried or nervous. Also a conversation or an event can be sticky. "Tough situation" is not necessarily embarrassing, it's just physically, mentally or emotionally hard to deal with. Something that requires a lot of effort and determination.

A sticky situation is an embarrassing one which needs to be handled delicately [ eg if an employee is caught stealing] whereas a tough situation is more serious for example whether to commit a country to war.