Will the Wither Fight the Ender Dragon?

The Wither can almost never hit the Ender Dragon, because despite the Wither's speed it will fly away too quickly, meaning the Wither's shots won't hit it. Once the Wither can no longer find the Ender Dragon, it will sink to the ground and get distracted by the endless Enderman minions.

Is the Wither harder than the Ender dragon?

The Wither is harder, plain and simple, because it can actually attack you. All the ender dragon does is attempt a ram, and is easily dissuaded.

Can we fight Ender dragon before Wither?

The wither is one of the two bosses in Minecraft, along with the ender dragon. It is usually considered the first boss of the game, even though it was added more recently and its advancements are unlocked after defeating the dragon.