Prior to clicking the shutter, the difference between professional-looking photos and snapshots is that the professional photos have taken a good number of aspects into consideration before taking a photo including, but not limited to:
- Ambient lighting
- Artificial/external lighting
- Composition
- Posing
- Background
- Foreground
- Depth of Field
- Clothing/Makeup
- Time of day
Snapshots typically do not consider all these aspects. They may consider some, not all. Professionals try to get the picture as good as possible in the camera, but the pictures usually do need to be processed to some extent after taking the picture. The processing is especially important if you're going for a certain look (e.g., really vibrant, matte, black and white).
After clicking the shutter, all photos are processed in some way. Professionals and enthusiasts usually shoot in RAW format and do the processing themselves with either paid or free software, depending on the photographer. Phone cameras and point and shoot cameras typically shoot in JPEG and let the camera do the processing for them.
If you're looking for better pictures using your camera (whether it's a P&S, phone, or a DSLR), you'll probably want to learn better composition, the exposure triangle, depth of field, and all about lighting (just to name a few). These items will elevate photos above the snapshot range.
Learning is fun and free after the purchase of the camera itself. With digital cameras you can take a picture, find out you don't like it, and then retake it. You can then process it on your computer using any number of free programs. You can also process photos using your smartphone with any number of free apps.
I agree with
@astroNikon, that you're probably trying to describe depth of field, where the subject is clear and the background is blurry. Learning depth of field is important, because it will help you get the look you're going for in your photos.