悔しいと煩わしいの違い何ですか。
How do you say "annoying" properly in japanese? Which is the most common way of saying it?

ありがとう！

"Annoying" means わずらわしい、うっとうしい, and other translations of わずらわしい are "troublesome", "complicated" depends on the things/situations. 

The meaning of 悔しい is regrettable. (悔しい思いをする = to feel frustrated / disappointed) 

Espero que te ayude.