Is Persona 4 Golden anime different from the original Persona 4 anime? Does Persona 4 Golden follow the same plot as the original Persona 4? I've seen the Persona 4 anime was wandering if Golden covered old ground. 

I know in the game there's a new character and/or party member but is this also reflected in the anime?
You have to watch Persona 4 before watching Persona 4 the Golden, the reason is in P4G they skip most of the plot presented in the original series and pretty much only animate the parts with the new character, Mary. So while watching the Golden you just have to fill in the missing parts with the original P4 or it won't make sense. Whatever happens in the original won't change except a small part in the end but even then it's not that much of a change, it's pretty much adding a final final true ending over the true ending of the original adaptation... Either that or play the Golden game for PS Vita since that one has the full story, not chopped up like the adaptation.

TL;DR P4G doesn't touch anything that happens in original P4. If anything, watching P4 priorhand is mandatory