Allison is right, unless you special order (and pay for) the birthday perks like a full size cake at a restaurant. (I don't know of any other special perks that WDW offers...except the free birthday buttons in the parks.)
I know this opens up a huge can of worms here on the DIS...almost as big as the refillable mug threads...

But, here is my share of thoughts on this...
Our DS has a mid May birthday, which is too close to final exams at the college DH teaches, so a family trip to WDW on DS's actual birthday is not possible for us. We have celebrated his birthday at WDW twice though...just a bit late (at the end of May and the first few days of June). It's not that far past his real birthdate so we see nothing wrong with it...now I don't think we'd go past a couple of weeks and celebrate a birthday, but I know some kids (of all ages) are lucky enough to get a trip to WDW for their birthday...DS's first trip was just this...so in all honesty, isn't that trip a celebration of the birthday despite when the person goes?
Also, there are many times that a person isn't able to celebrate his/her birthday on the exact date of birth even when they just celebrate at home. I mean, my grandma never was able to really celebrate her birthday on the day she was born...December 25. 
And I don't remember too many birthday parties for anyone ever being on the exact birthday, but rather on the nearest weekend and usually on the Saturday.
So, what I'm saying is...if you want to celebrate your birthday at WDW and it happens to be a little past the exact date of your birth...then go for it!

And enjoy! Get the birthday buttons from all the parks and at DTD, and hey, if you are inclined to mention you are celebrating your birthday when you make any ADRs and you get a little treat...say thanks and enjoy! And for anyone celebrating a bithday in WDW, on the exact date or not...if you don't get anything special, well, it's still a magical day.
For pete's sake...when I had to make a special order for the Cinderella's Slipper Dessert at CRT for our Christmas trip, the CM asked if I was celebrating any special occasion. I said..."uh, well, Christmas at WDW isn't that enough?" The CM chuckled. I didn't think anything of it until my slipper arrived a month later during our meal and it said "Merry Christmas Laura" on the plate...guess WDW celebrates about anything.
Now, having said all this, I am prepared to take the flames which I suspect may follow...but to each his own. It's your trip...celebrate and enjoy it as you see fit!