Finnish birthday traditions are pretty similar to any other western culture: cards, presents, cake, blowing the candles, singing Finnish version of Happy Birthday Song... It's actually quite surprising that the popular Swedish Birthday Song hasn't been translated to Finnish, and hence it's not so popular in Finland (or at least I'm not aware of it). Well, my favorite birthday song is anyway coming from the other neighbor, Russia: Crocodile Gena's Birthday Song, and the excellent Finnish version of the song.
The impact of Anglo-Saxon culture is clearly present on the birthday wishes. Still in the 80's people were using more of the traditional Finnish birthday wish onneksi olkoon - let there be luck. Nowadays people are often taking the direct translation from English: hyvää syntymäpäivää - Happy Birthday.