like, is it two separate languages that are just common above/under ground respectively, or is one a offshoot of the other, like Australian or American English is a off shoot of British english?

The first one.  "Common" and "Undercommon" are terms of convenience to describe what are linguistically known as pidgins, although Common has probably evolved into a creole, since it has native speakers.  Undercommon is a simplified trade language to let different groups communicate in the underdark.  Drow, Duergar, Svirfniblin, Dwarven, Kuo-Toa...when they need to communicate with one another, they use Undercommon.  On the other hand, when they go home to Citadel Adbar and Menzo, they speak Dwarvish and Elvish to their family.

Common started off something like that, but evolved to apparently take the place of any other human language in the Realms at least, making it a stable language in its own right.  As far as anyone has ever written down, Undercommon shares neither grammar, nor vocabulary, nor a culture of native speakers, with Common.  The only similarity is their purpose.

In my world, Common is quite simply the most common of Mannish tongues, in a similar way to Latin or Greek during and immediately after the Roman empire. (Man being a term used to describe Humankind, which includes halflings, tieflings, aasimar, and about half of all half-elves [just like in Tolkien, a half-elf has to choose between being classed as a Man or an Elf]). Undercommon, on the other hand, is a pidgin language that started as a bad grammar version of the drow dialect of Elvish, and gradually got more and more debased (mainly in the hands of the Duergar), such that it is not mutually comprehensible with the Drow dialect. There are a few native speakers of Undercommon, these are generally a mixture of those who are born into slavery to the drow or the duergar, or those who are born into one of the various merchant companies of the underdark.

I like Tim's explanation interpretation, but Common in the PHB is simply the Human language; but to support Tim and do him one over, I would say Common isn't necessarily a pidgin or creole, but definitely a Lingua Franca, the wikipedia description of a Lingua Franca in function is basically what Common allows in game worlds.

Also, yeah, no relationship between common and under common.  Undercommon is just the Lingua Franca of the Underdark.  In the PHB Common is listed as a standard language and Undercommon an exotic language.  If your game was Underdark based, the situation would flip.

So is Deep Speech the Lingua Franca of Aberations?  Do Beholders and Mindflayers and Aboleths really get together for conversations much?  Like maybe a standing poker game or maybe a full blown convention with some guest speakers from the Far Realm?

Language isn't sophisticated in D&D mechanically.  You either know it or not, and have magical aids if you don't and need to.  It's easily ignored unless the DM has reasons to make it a thing.