They're just nice little animations, nothing really special. Some do make your Tamagotchi beep though so be aware of that.
Some animations are exclusive to some versions, like some only seem to appear on v1 and v2. Note that I live in Australia so I find some of the seasonal events pretty strange and our seasons come at different times (it's Spring now, almost Summer)
When your Tamagotchi wakes up on your birthday it'll celebrate your birthday.
At midnight on New Year's Eve your Tamagotchi will celebrate New Year's Eve (unless it's as glitchy as my v3- it didn't celebrate New Year's Eve on the right day, but it celebrated it three times the next day!)
At 11pm (I think) on Christmas Eve Santaclautchi comes to your Tamagotchi.
During Thanksgiving a turkey appears.
During Autumn (which is Spring for most of you) your Tamagotchi plants flowers.
During September-ish your Tamagotchi wears a backpack.
During Summer (which is Winter for most of you) a snowman appears.
During Winter (which is Summer for most of you) your Tamagotchi goes to the beach.
During Halloween a pumpkin appears.
During December a Christmas tree appears.
I'm pretty sure I've left some out but that's all that I can think of.
