Robots are generally under things that are automated, mechs are more usually human controlled... In my experience, more like a mechanical augmentation than a stand alone robotic.
Although i'm sure you'll drag up some japanese example that disproves this, then again the definitions
ROBOT
1 a : a machine that looks like a human being and performs various complex acts (as walking or talking) of a human being; also : a similar but fictional machine whose lack of capacity for human emotions is often emphasized b : an efficient insensitive person who functions automatically
2 : a device that automatically performs complicated often repetitive tasks
3 : a mechanism guided by automatic controls
note: the word auto
The word mechis much more vague, since 'mechs' are simply a sci-fi concept there's no definition for the word, other than that it is an abbreviation of the word mechanic, which is a noun and adjective,
MECHANIC (adjective)
1 : of or relating to manual work or skill
2 : MECHANICAL
MECHANIC (noun)
1 : a manual worker : ARTISAN
2 : MACHINIST; especially : one who repairs machines
MECHANICAL
1 a (1) : of or relating to machinery or tools <mechanical applications of science> <a mechanical genius> <mechanical aptitude> (2) : produced or operated by a machine or tool <mechanical power> <a mechanical refrigerator> <a mechanical saw> b : of or relating to manual operations
2 : of or relating to artisans or machinists <the mechanical trades>
3 a : done as if by machine : seemingly uninfluenced by the mind or emotions : AUTOMATIC <her singing was cold and mechanical> b : of or relating to technicalities or petty matters
4 a : relating to, governed by, or in accordance with the principles of mechanics <mechanical work> <mechanical energy> b : relating to the quantitative relations of force and matter <mechanical pressure of wind on a tower>
5 : caused by, resulting from, or relating to a process that involves a purely physical as opposed to a chemical change <mechanical erosion of rock>
By any means, the sci-fi definition appears to be one of a living creature using a mechanical suit or vehicle to enhance combat.
If that defines the lines, I'll be pleased.
note: these definitions are from the merriam-webster dictionary, an American-English dictionary, so before anyone corrects me on spelling mistakes or points out then it's not the English definitions, I tend to be believe Americans have the jurisdiction on sci-fi these days
