What's the difference?
Why, savings to you the consumer!
In the days of the olde, everything was forged and then milled until it conformed to a specified standard that was generally measured with jigs or "go-gauges." Advantage of forging is that under all that hammer, you align the molecules giving the metal greater strength. Disadvantage - higher labor costs (both in terms of forging and subsequent machining) and greater waste (a lot of the forging ends up as filing or metal chips).
Casting is actually a very ancient process and was used for jewelry making thousands of years ago. Originally, one took the object they wished to duplicate, placed it into sand (or clay), packed it tightly and then removed it. Hot metal was poured into the clay and when the metal cooled and hardened, the clay removed leaving a rough casting (rough depending on the clay's quality). Another method was to pack the object around sand and after the object was removed, the metal was poured (hence sand casting). Lost wax was a process whereby a wax duplicate of the object you wanted to mold was created and then clay packed around it. The clay would be heated, melting the wax and leaving a "reverse" image in the clay. The molten metal was then poured in and after it cooled, the hardened clay removed to yield the casting. It wasn't until perhaps the late '40s or early '50s when Springfield Armory (yep, the real government armory) experimented with castings. Bill Ruger (Pine Tree Castings) made casting big and the advantages are you can make intricate parts that would be otherwise difficult to machine relatively cheaply. Not that they don't require some machining, but it cost less. This freed the engineers from design constraints limited by machine manufacturing (and costs).
Oh, as to the disadvantages, the porosity of the metal makes it more suspectible to failure. A large part of this though depends on the quality of the foundry doing the casting. Ruger does an excellant job while others, well, make junk. Major advantage of casting is reduced labor expense and reduce metal wastage (sprue is tossed back into the smelting pot).
Now stamping was something that saw extensive use during WW II by both the Germans (MG-42, MP-40 smg, Stgw 44), Russians (PPsh submachinegun), Americans (M-3 Grease Gun, Liberator and even an experimental 1911) and the British (Sten gun had stampings I think). As described above, one took sheet metal and placed it into a press that had dies (machined) around which the metal was cut and formed. The advantage of stamping is that parts could be made very cheaply but were not "cheap" in terms of durability. Many modern pistols utilize stamped components including Glock and Sig Sauer. Advantage: inexpensive. Disadvantage - some consider it unsightly. Want to see a major stamping? Look at the Sig P220 or P226 slide. Those are stamped. The P229 or P239 are casted (by Ruger) and then machined (by Sig).