First thing would be, what do you want to use it for?
I'll do my best to outline the differences between them and leave it to you decide which one you want. all comparisons are 'pound for pound', i.e. all things being equal other than calibre.
.177 has a flatter trajectory as it's a smaller, lighter pellet and travels much faster than .22. This makes it more forgiving from a marksmanship point of view as range estimation isn't as critical. This is due to less holdover/ holdunder being required when aiming at different ranges, if you're a few yards out with your range estimation it won't make too much difference. As it travels faster it'll get to it's target quicker too.
A smaller faster projectile will pass through and soft target (animal) easier, when this happens it means it's not delivering all of it's force into the target.
.22 is basically the opposite of the above. .22 rarely passes through an animal giving rise to the phrase "more stopping power", but the truth is either are fine for hunting as if you put the pellet in the right place it'll make the kill. it's true a .22 "hits harder" but you have to be on top of your marksmanship to be as accurate as a .177, especially when shooting spring piston rifles.
.177 pellets are cheaper.
for target shooting most people use .177 for the flatter trajectory and smaller pellet holes (meaning you can fit more individual holes in a target card). For hunting there is much debate!! but as I said either does fine and I hunt with both with success.
hope this helps.
Jim