The first thing I noticed was the beauty of being able to control depth of field! (how much of the picture is in focus) This is an option you don't have with most Point and Shoot cameras. Manufacturers know that most people, just taking snapshots, want everything in focus. (Or, they "think" they want everything in focus.) Because of this they program the camera to "make it so!" Not so the DSLR! Oh, no! It will gladly blur any part of the picture that you desire and sometimes parts that you don't desire. But, oh, the beauty when you learn to use this seemingly undesirable trait!
Notice the picture above of this ordinary little ball of fluff. Now take a look at the background. Did you notice that it is out of focus but still it does not detract from the picture but rather, enhances it. The sharp foreground draws your eye to the subject of your photograph while the background does just what it is supposed to do...be a background. That is actually the way we see things in real life. Think about it - when you focus on an object, only that one thing is in focus. The rest is a blur until you move your eyes!
Here is another little "jewel" with a small depth of field.
Don't take my word for it! Beg, borrow, (don't) steal a DSLR and try it for yourself. You will never look back!