How are these affects different?

Legolas: "After Legolas participates in an attack that destroys an enemy, place 2 progress tokens on the current quest."

Brand: "After Brand son of Bain attacks and defeats an enemy engaged with another player, choose and ready one of that player's characters."

What does the word "participate" imply? Also, the Blade of Gondolin has wording similar to Brand's.

Does this mean Legolas can be part of a group-attack that destroys an enemy and still trigger his response, whereas Brand has to single-handedly defeat the enemy to trigger his response?

If a character is part of an attack then they participate. Yes, Legolas can use his ranged ability to destroy enemies engaged with other players and his Response is triggered (that's why it's called a "response"). If he has the Blade of Gondolin attached then you get to place an extra progress token (3 instead of 2).

No, Brand does not have to single-handedly attack and kill an enemy to trigger his ability. He simply has to be a partcipant.

Thanks. I'm glad to hear that.