Both Ted and Teddy work as nicknames for Thaddeus. I prefer Ted. It reminds me of a really cool Ted who was my mentor once upon a time. Tad reminds me of Abraham Lincoln's son. Tad is OK, though -- it's just that I prefer Ted.
You didn't ask, but I don't care much for Thaddeus. It seems on the pretentious side (now that I've said that, I'm really hoping it isn't the name of a grandson -- I can't remember their names!). My apologies if Thaddeus is one of your grandsons.