I think it really comes down to the seriousness with which you treat your photography. If you're going to go picture taking with the primary objective of taking pictures, then you should get a DSLR. If you want to casually enjoy your picture taking opportunities and only occasionally take pictures, then go with a digital compact. The most obvious reason is bulk and the next reason is cost.
DSLRs have large bodies that do not fit casually into a pocket, so if you are going somewhere with your camera you will have to be ready to carry a large camera and accompanying lens for the duration of your journey. A compact digital camera is not very heavy and most can easily be slipped into a bag or pocket. You may not even realize you have your digital compact with you until you think to take a picture.
As for cost, unless money is easy to come by for you, you will notice the significant difference in the price tag. If you like to take pictures in environments that may adversely affect your camera, the risk of damaging your expensive camera may inhibit your willingness to take pictures. However, with a digital compact the potential cost of damage is much smaller, so you may be willing to take it to the beach, or a child's birthday party, or in the rain.
Yes, there is a significant difference in picture quality. There is also a difference in suitable subjects. DSLRs have the potential to create magazine quality photos of subjects that may not be possible with a digital compact due to their versatility. However, you must decide based on the other criteria first if these are prices you're willing to pay for your pictures.
Whatever you decide, you can greatly improve the quality of pictures from either your DSLR or your digital compact by investing in photoshop and digitally enhancing your photos there. That's what all the professionals use. Your photo enhancement software can often mean the difference between a dull photo and a lively one.