When I use a pickaxe under water it takes longer. If I continue to work under water will my pick break faster than if I used it above water? I would like to know if it is worth all the time, and I think it would be if it broke at the same speed.
It will last just as long.
Tools will not break faster if used underwater. There is no difference in the durability expended between being submerged or dry.
Tools will, however, take longer to break blocks than they would if you were on land. This has nothing to do with durability - durability is calculated per block broken rather than time spent mining.
See Item Durability on the Minecraft Wiki.
Your pickaxe will have exactly the same durability. The durability is calculated by the number of blocks mined, not by the time spent mining.
Your durability decreases by 1 per block mined. (Unless you use the Unbreaking enchantment).
So it doesn't matter if you mine at surface or underwater, or if you mine dirt or obsidian, the durability will always decrease by 1 point. (Unless, see above.)