Deep discount fares are typically more restricted fares that are cheaper than simply discount fares.

For example...
LAX-LGA round trip fare on AA leaving May 5, returning May 8

Cheapest deep discount fare: $238++
Cheapest discount fare: $558++

And before you say that you don't think the $558++ is a discount fare, the full regular fare for this example is $3148.

All fares on AA earn full mileage but the deep discount fares typically only receive 0.5 elite qualifying points per mile earned. See this.

I don't fly delta and am not familiar with their program although the general information and idea here is the same.