That's a huge question! Welcome to DB Scoles
Well, I would recommend you find past threads on bi-fins and monos, but to give my two cents:
Monofins:
Pros:
-great for going in a straight line.
-very efficient once you learn the proper technique.
-most of the top freedivers these days all use monos for the deepest constant weight dives and almost the longest dynamics.
-spread lactic acid over a larger range of muscle groups as almost the entire body is used while undulating for propulsion.
-can reach great speed with proper technique.
Cons:
-depending on the kind of footpocket (i.e.- open heel or bi-fin pocket like OMER), the open heel pockets can be uncomfortable when worn for an extended period time.
-more inconvenient to use a mono for recreational all day diving compared to
bi-fins.
-usually more expensive than bi-fins (about the same when you get into carbon fiber / exotics though).
-harder to learn proper form.
Bi-Fins:
Pros:
-great for learning how to freedive and for recreational use-although many compete and set records with bi-fins as well.
-perfect for spearfishing.
-easier to learn generally, more natural.
-usually less expensive and easier to acquire than monofins.
-lots of different models and footpockets out there - easy to shop around.
Cons:
-not as efficient or fast as a mono which may ultimately limit your max depth -i.e. - constant weight.
-easy to develop wrong kicking technique if starting out without coaching - i.e. - bicycle kick, excessive knee bend.
I'm sure others on DB can contribute, but this is just to get you started, and of course, this is just my humble opinion.