Welcome pencil popper. I'm not and expert long distance surfcaster but since my fishing background is similar to yours....predominately a Large mouth fisherman for many years before targeting stripers.....I can relate and perhaps give you some insight as to the differences.
The biggest difference I find is the way you "load" the rod, the backward bend of the pole which provides the power/speed that allows you to cast long distances. This is because LMB equipment is much lighter and smaller, this includes the terminal tackle as well. With the lighter tackle, I depend primarily on the back-cast to load the rod. With surfcasting equipment, loading the rod is done primarily with the forward movement of the rod.
To effectively load the surfcasting rod, begin the forward cast slowly. Then smoothly accelerate by pushing with your right hand and pulling on the rod butt with your left (opposite if left handed). To reach maximum speed, break your wrists just before you stop the rods forward movement and release the line. To use all the power stored in a loaded rod, you must abruptly stop the rod without lowering the tip from the target line of travel, about a 33? angle, or 11 o'clock, from the waters surface).
The second biggest difference with the surfcasting technique is the exaggerated body dynamics. To use Sudsy's golf analogy, let's call the action of the rod described above, the "Stroke". And, let's draw on another sport, baseball, to describe the specific body dynamics I'll call the "windup". With lighter weight freshwater equipment, your stroke is much shorter and your windup is primarily all arm and wrist with hardly any need for additional body movement/augmentation. Whereas, with surfcasting you use your whole body by winding up or coiling to increase your stroke and provide more speed which translates into distance. BTW, the reason surfcasting rods are so long is basically to increase the length of your stroke.
I tried to explain in detail my body dynamics in setting up to cast as well as the cast itself but, I erased it because after writing it I had difficulty understanding what I wrote.
So I'll try to keep it simple.
First, let me say that your body dynamics are meant to work in sync/parallel with your rod movement to increase the length and speed of your stroke. To do this, your stance is very important to achieving a long-distance cast. Being right-handed, my left foot is aimed slightly right of the spot I'm casting to and my right foot is at slightly less than a 45? angle to my left foot. Both feet are spread about a shoulders-width apart. Start with your plug hanging about 18-24 inches positioned below your rod tip. When you bring your rod back to the casting position (~horizontal to the water, except obviously pointed away from the water) you want to rotate your hips to the right as well as shift your body weight to your right foot. Here, you're effectively loading your body. Keeping in mind the rod dynamics that I explained above, as you begin your cast, push off with your right foot, slowly shift your weight forward (right to left foot) and begin to rotate your hips back to the left. This windup, body weight shift and hip movement, augments the strokes length and provides additional speed.
Now, put the mechanics I've described regarding the stroke and windup together and you should get decent casting distance. Obviously, the plug's weight and aerodynamics will impact your distance and it takes practice to become consistent.
I hope you can visualize what I'm describing. In this case, an instruction video is definately worth a thousand written words. Oh well, I tried.
Note on edit: added the baseball pitchers windup as an analogy to help describle your body movement dynamics.