In a nutshell, a DIN regulator is screwed into a DIN valve (like a bolt into a nut), whereas a yoke regulator is clamped over a yoke valve, relying partly on air pressure to keep it secure. Yoke valves are commonly (rightly or wrongly) refered to as K valves.
DIN is a more secure fitting, slightly more compact, can handle higher working pressure tanks than yoke, and has a reduced likelyhood of a blown o-ring ruining your dive. Internally, DIN and yoke regulators do not differ, and depending on the manufacturer, can often be converted from one type to the other.
I'm moving from yoke to DIN in the very near future as I think it is a superior system. It is very popular outside of the United States, particularly in Europe. Yoke is historically more popular in the US.
A regulator purchase decision regarding yoke or DIN should be based on if you own yoke or DIN tanks, or the availability of yoke or DIN rental tanks. Travel adapters can adapt a DIN regulator to yoke if travelling to a destination where DIN is not available.