I thought he was called Blue in the manga. Green is a female in the manga. Gary is the adaptation of Blue in the anime.
Hold your horses. That depends entirely on in what language or what translation of the Pokémon Adventures manga you have read. In that manga, I have always called Red's rival Green, since that's his original Japanese name while the girl is Blue.
To answer the original topic, yes there's a difference. If hardcore fans get angry about you posting pics of Gary on Green's birthday, then it's the male rival in that manga they are referring to. I don't think anyone keeps track of Gary/Blue's birthday in the games :p
Basically, there's three well known versions of him:
- Gary (anime)
- Blue/Gary (English games)
- Green/Blue (Pokémon Special Manga, he's called Blue in the official English translations methinks)
And since game Blue/Gary is endlessly less cool than manga Green, I would also raise an eyebrow if someone mixed them up ;) I wouldn't be angry though, I'd just point it out, haha. It's usually easy to recognize anime Gary since his hair is larger than any of the others and he sports an Umbreon or Eevee (?). Manga Green has a Charizard or Scyther/Scizor, often, and game Blue/Gary is often portrayed with an Arcanine or Pidgeot, from what I've seen.
The pic you posted looks mostly like game Blue/Gary or possibly anime Gary since 1) why would tough Green be smiling like that with loads of Eevee? 2) his eyes aren't green :)