Since there's a concept page for each, I think the blurb for each should specify when somethings a robot and when something's a mech, otherwise combine the pages.

a robot is something that has some sort of artificial intelligence... a mech must be piloted by somebody.

" a robot is something that has some sort of artificial intelligence... a mech must be piloted by somebody. "

Mech is a term generally related to Varitechs, Gundams, Mazinger, etc. Machines that require pilots.
Robot/Androids is a term related so self suficient machines, like Megaman, or R2-D2. 

There's also an object page for Mech. I think their blurbs are quite different:

Robot: "A mechanical lifeform, typically exhibiting superhuman abilities, that can be found in games as either the main protaganist or as powerful enemies or allies."

Mech: "They must be big, they must have legs (but not always), and they must have cockpits. That's a Mech!"

While I don't really agree with the tone of the Mech's blurb, the two do show a clear difference.

Edit: I bolded the more important differences.

" a robot is something that has some sort of artificial intelligence... a mech must be piloted by somebody. "

Totally agree:

Robots are autonomous, while mechs are not.

Wait. What about remote-control robots?

Mech normally just refers to bigness. Plenty of mechs have no pilot. Under 10 feet tall, its a robot. Over 10 feet tall, its a mech. There.

robot autoaamtion that acts on its own (R2-D2)
 
Mech- must be piloted like a gundam

A ton of mechs in the Gundam series have no pilot. It normally serves as a plot twist or something, "oh no they made mechs that don't need pilots". The enemy mechs in Gurren Lagan have no pilot. NGE mechs function without a pilot at certain points. etc etc other nerdy references I'm ashamed of.

I think 'robot' simply (and, as a result, quite broadly) refers to any complex practical machine designed to operate or be operated remotely to perform a specific function. A bomb disposal robot, the robots used in car factories and things like R2D2 are 'robots'. Furthermore, robots that are specifically humanoid and largely independent are androids, such as C3P0 or Megaman. The designation of 'mech' or 'mecha' to describe some kind of mechanical personal armour (unmanned or otherwise) should largely be up to the intention of the author/ designer, however they are still robots. I don't think a genuinely accurate description can be made, though it's most appropriate in media either from or influenced by Japanese games/ manga/ animation. The term serves well enough to cover the broad range of walking tanks/ power armour/ 'gundam' type units you'll see, but it's certainly not clear. Is a hound from Chromehounds a mech? If yes, does that make an AT-ST from Star Wars a mech too, seen as both are non-humanoid, piloted combat walkers?  
 
EDIT: Also, damn, didn't realise this thread was fully necro'd. Nothing quite like giving your two bits to a discussion that happened a year and a half ago...

" A ton of mechs in the Gundam series have no pilot. It normally serves as a plot twist or something, "oh no they made mechs that don't need pilots". The enemy mechs in Gurren Lagan have no pilot. NGE mechs function without a pilot at certain points. etc etc other nerdy references I'm ashamed of. "