What is the definition of a "pug"?

They are the same thing. Folks that come from a MMORPG lineage, especially WoW, will tend to call them PUGs (Pick up groups). PUG is also defined as "Pick up game" in some circles. Folks who haven't been in the MMORPG scene may simply refer to them as Pub, short for Public Groups.

All told, the common vein is that these are games with people you don't know. They are called out in discussion usually because of the negative connotation of choosing to mix with the unwashed masses who make up the internet at large. The "Online Disinhibition Effect" (which you may know by it's NSFW name) tends to exhibit heavily in games of this type, resulting in a negative experience for those that just want to play the game.

"'pugs' are instead games with random people on the internet in a competitive format." - badp, What is the definition of a "pug"? .
PUGs are not merely public games, as badp mentions. YouTube's DrMattDestruction has a video series called "Counter-Strike Minute". This demonstrates well the fusion of random players (public) with competitive format, particular to PUGs. It's clear that it's not unified clan members competing or scrimmaging against one another, but random players. And there's a minimum of skill expected from other players.

"With [their] pug system that allows you to play with the pros, whether you are a new comer to the game, or an experienced player." - http://en.wikipedia.org/wiki/ESEA_League .
This is the "league" in which the games of "Counter-Strike Minute" take place. It has subscription-based access, which can allow it to keep a higher standard. "pug system" implies a sense of order.

I assume that PUG, as an acronym, can be broken down like this: PU is for public or pick-up, and G is for game or group. Abbreviation "pub": short for public. Abbreviation of public game, group, or server. Can be used as a verb, as in "I was pubbing on zombie servers" which means "I was playing in servers that are open to the public, which host zombie game modes."
"(Pick-up) Game", in PUG, can emphasize the competitive aspect more than "public server".
Pubstomp: verb (intransitive), portmanteau, meaning "play very well in public servers" if not "sufficiently disrupt the gameplay experience of others with the intent to do so".

While PUG (pick-up game/group) and pub (public game/server) may be interchangeable in MMORPGs, it is uncommon for PUG to be interchangeable with pub in FPS.