Events guarantee bonuses like more gold from the campaign or cheaper Gear at the Item Shop which can be found in the Grand Event zeppelin (that flying ship!).

They also give you a chance to get the newest Gear sets from the Event Chests, and some extra rewards from Top-Up/Spend competitions. A new Event starts on Mondays at 17:05 UTC, and Wednesdays at 13:00 UTC.

Battle Event is a special competition where you're fighting for valuable prizes by collecting Trophies. Trophies drop from Battle Event Campaign and Battle Event Bosses.
The most important element of Battle Events mechanics is the Multiplier.

Every Battle Event features a different Emblem - which is a symbol describing a family of Gear. Battle Events starts on Friday at 13:00 UTC and ends on Mondays at 16:00 UTC.