A regular clean is designed to help maintain a certain level of cleanliness around your house such as vacuuming, dusting, mopping, cleaning sinks, toilets, etc.
A deep cleaning service will remove the deep dirt and grime in your home. It will cover the areas that aren't typically covered in a regular cleaning service and take care of those spots that are hard to clean or are rarely cleaned such as removing scale and soap scum from shower heads, cleaning behind appliances, scrubbing baseboards, etc.