The difference between tai and vai

Both tai and vai mean "or"

1. Tai

When using tai, you are allowing your listener to pick whatever number of possibilites they would like.

Finnish English
Haluatko sokeria tai kermaa? Do you want sugar or cream?
The listener can take cream, sugar, both or neither
Isä, saanko jäätelöä tai suklaata? Dad, can I have icecream or chocolate?
Dad can say icrecream, chocolate, both or neither

2. Vai

You are making your listener choose between the possiblities you offer.

Finnish English
Haluatko kahvia vai teetä? Do you want coffee or tea?
The listener can take coffee, or tea. You are ruling out that he/she takes both of them.
Kumpi on vanhempi kaupunki, Tampere vai Turku? Which is the older city, Tampere or Turku?
The listener can either pick Tampere or Turku, but not both.

Vai also has a second meaning: the beginning of a short (not very enthousiastic) reaction to something said, asking for a confirmation.

Finnish English
Vai niin. Oh, like that.
Vai teetä. So, tea.
Vai mitä? Right? Isn't it?