I'm not quite sure how to word this well, so please forgive the aukward wording.
It shows more consideration to separate gifts for a single person by occasion instead of time of the year. By this I mean people with December birthdays might not really want a Christmas themed item (Santa shirt for example) given to them as their birthday gift, while that exact same gift would have been much appreciated at Christmas.
Corollory to this: People with December birthdays really do appreciate the effort if you DON'T wrap their birthday presents in Christmas paper because "it was to hand". December folks have to find non-seasonal paper for the rest of you, don't we deserve the same courtesy?!
Also: When giving gifts to a December birthday person (particularly someone with a birthday very close to Christmas) it's an extremely nice gesture to recognise the two occasions separately rather than saying "Oh, it's a joint Christmas/birthday present." See above regarding December people and other folks' birthdays...
[NB This last part is void if the gift is something that would be a massively inflated single present (eg the year I got my clarinet, which was a joint birthday/Christmas present). If funds do not permit a gift for both occasions, just call it a Christmas gift and call it done!]
Anyone want to hazard a guess when my birthday is?...!
ETA (because I realise that sounds a little "gimme") It's not about getting lots of presents (or even getting ANY presents!), it's more about people recognising that December Birthday =/= Christmas and Christmas =/= December Birthday (unless you're talking the relevant religious birthday, which is a topic for elsewhere entirely)