No reason why you couldn't reuse the 2600k cooler, its the same socket, and rated to cool a 95w processor, vrs the stock 3770k which is 77w, so it'll be good.
Aftermarket cooling is easy to answer. Decent coolers have a much greater cooling capability vrs stock. Your stock cooler will handle @100w worth of heat at max before it allows temps to rise beyond the processors limits. Even a budget cooler such as the CoolerMaster hyper212 EVO has a TDP of 180w, so at stock speeds the fan will never reach over 70% speeds and still keep the cpu well under thermal limits.
This is why aftermarket coolers are a must for any OC, as the cpu will be working harder, so getting hotter than a stock cooler can handle safely.
The other reason is noise. With a stock fan at max during a game, or watching a movie, it simply gets annoying. With much slower fan speeds capable with better cooling, its much! quieter, more enjoyable.
Liquid all in one's (aio's or clc's) have strengths and weaknesses, as do air coolers. Clc's tend to cost more, but are ultimately better than the best aircoolers. Not everybody needs that much cooling though. Clc's will fit in any case with a 120mm or 140mm Fan slot, air coolers are restricted by height in a lot of smaller cases. Clc's have no ram clearance issues, large airs do. Clc's can have an annoying HP in the pump, no pump on an air cooler. The possibility exists of a coolant leak in a clc, possibility also exists of warping or cracking a motherboard with a top heavy aircooler.
Some ppl simply prefer the clean pc look of a clc, some prefer the muscle car look of a big tower.
It all depends on your budget, your aesthetics, needs and wants