Both are fantastic to visit for different reasons.
Europe is undeniably glorious for its inexhaustible amount of culture and history. There isn’t a place in the world that has more staggering art, architecture, culture and historical things to see and do. Even if you decide to go to Asia first, you must visit Europe in your lifetime.
Asia is huge and very diverse, so I’m assuming you’re referring to South East Asia because that’s where most backpackers go. It is a lot cheaper than Europe, and it will probably lead to a more adventurous journey. There are fewer rules, and things are a bit crazier and different than what you’re used to (assuming you live in a Western nation).
Traveling through Thailand and Bali are definitely on my list of unforgettable experiences. I’ll never forget riding a motorcycle through the chaotic traffic over there, eating street-food BBQ, walking through night markets and eating strange things, attending a Muay Thai training with locals and later sitting ringside at a Muay Thai fight (for $15), and watching a gorgeous sunset over Koh Phangan.
But so do driving down the Autobahn at 250km/h, marveling at some of the most incredible paintings ever made in the French Louvre and the Dutch Rijksmuseum, walking through mind-blowing cities like Venice, Paris, and Bruges, and drinking beer in a Belgian monastery brewed by monks for hundreds of years.
The world is an interesting place. Try to see as much as you can.