At first glance they don't seem to stylistically similar (different stroke weight, ascenders and descenders), but I'm not terribly knowledgeable myself so some constructive input would be great.

IF YOU LOOK WITH YOUR EYES THEY ARE DIFFERENT AS SKIING AND BREASTFEEDING BABY.

IF YOU ARE NOT KNOWLEDGEABLE YOU SHOULD NOT SAY EH? SAD. I'M SURE YOU DO BETTER IN FUTURE.

Galaxie Copernicus is a take on Plantin, a 16th-century genre, and Georgia is a Scotch Roman, a 19th-century genre. Plantin does have a large x-height compared to other designs of its era, but if you go into the details, there is very little similarity between the two genres and hence the two faces.

However, Georgia, as you noted, was designed for the screen, and Galaxie Copernicus was designed as a member of the Galaxie family that includes a grotesque (a sans serif subgenre), so those decisions probably affected the proportions, colour, and sturdiness of the designs. That could be said about a lot of other contemporary designs, though.

To answer your question in the title: Not much. They’re both serif typefaces.
But Georgia is a Scotch Roman optimized for the screen, whereas Copernicus is a typeface stylisticly similar to Plantin, a French Old Style face.

Check this: Jonathan Hoefler’s Type Styles

 
Copernicus above, Georgia below

Quite different, but certainly more similar than Monotype's early 20th century revivals.
From Stanley Morison's A Tally of Types:

Nick, do you mean the Scotch Roman or the Plantin in your image?

Both.

:-)

I bought Copernicus in one weight not too long ago.
I bought it because it was markedly different from Georgia and I wanted to test it out on screen as an alternate body font. (Won't get into the results of that here.)
The biggest diff I notice right off was the counters in the letters B, D, O, and the like. Much larger than Georgia.

My eye is anything but trained with regards to typefaces.

Forgive my being flip, but have you been to the optometrist lately? ;)