I've written quite a bit here on why you should consider spinning for trout in streams, but I don't think I've highlighted fully the differences between fishing with a fly rod and fishing with a spinning rod.
The most basic difference between fly fishing and spinning or "regular" fishing is that when you use a spinning rod and reel you depend on the weight of the lure (or sinker if fishing with bait) to cast, but when fishing with a fly rod you are depending on the weight of the line to make your cast.
So the motions might initially look similar, but they are in fact, quite different.
When fly casting, you start out with a few feet of line off the end of the rod tip and perhaps several feet of line between the reel and the first eyelet of the rod. The cast begins with your rod in front of you, you lift the line up and behind you. When the line is fully extended behind you, you deliberately move the rod tip forward, casting the line out on the water, letting some of the line that you had pulled out of the reel out the end of your rod. It may actually take several casts for you to get enough line out to reach your target. In fly casting, the only function of your reel is to store your line.
When spin casting, your casting motion really begins with your rod behind you, with the lure a few inches from the tip of your rod. You open the bail on your reel (which will allow line out), but hold the line so it doesn't escape from the reel. As you whip your rod forward, you release the line allowing line to escape and the lure to reach its target. In one cast. The function of a spinning reel is to store line, but also to control tension when reeling in a catch (through the drag) and in the case of a spincasting reel (which has an enclosed reel with a button - likely the first reel you ever fished with as a child) will release the line on command to assist in the cast.
Through this description, you can see how fly fishing is a sport of finesse. Your motions are deliberate and controlled. If you move your rod too quickly you're likely to snap your leader and lose your fly. If you move it too slowly, the cast won't go anywhere.
Spinning on the other hand has considerable room for error. And you should error on side of too much power. Anyone who has casted a spinning rod will tell you that a larger lure means a longer cast (until you start into extremes) a little extra weight will pull more line out and travel through the air further. There's not a lot of finesse, but it helps if you have good aim.
Another difference is once you've hooked your catch. A fly rod depends largely on the flex of the rod and the actions of the fisherman to maintain the right amount of tension on the line. Too much tension will cause the line to break, and too little will allow the fish to spit the hook out of its mouth. Either way you've lost the fish. Spinning rods also rely on the flex of the rod and the actions of the fisherman, but tension is also controlled by the drag of the reel - again, giving the fisherman greater room for error. If there's too much tension on the line, the reel will start letting out line, preventing the line from breaking.
If you have the time and desire to master an art, fly fishing may be for you. It takes months to learn and years to master, but the rewards for doing so are great. On the other hand, if you want to be able to just go fishing, have fun and catch some fish, you may want to bookmark SpinnerFishingforTrout.com - because spinning for trout is for you!