Hey Jamie, This is a semi dynamic question and I will try to lay out the answers you are looking for in a simple way.

Intially,.let's compare a slick and a DOT of the SAME COMPOUND.

First let's look at DOTs:
-The defining factor that classifies a tire as a DOT is that the overall surface area needs to have a void of 25% or more than that of a slick tire. This is the tread pattern you see that each brand/model has (plus other underlying factors). Where the void are placed is up to the manufacturer. (Notice that most continual rubber is on the sides of the tire.)
So this means that less surface area (on average) will be on the track with a DOT than a slick.
-The void of rubber that the tread pattern creates also leaves less mass on a DOT tire compared to a slick. (you would think it should be 25% less or more because that's wants missing from the overall surface area but that not true because those tread patterns don't extend all the way beyond the carcass.) However, the less material (rubber) and mass makes the DOT a lot easier to heat up and to cool down. Think of warming up and cooling down a peice of aluminum foil as opposed to a brick. So this means that a DOT has a much wider range of a usable temperature window.

As for the slick:
The inverse is true.
-Has more grip than a DOT
-smaller usable temperature window.
- You would think that since the slick has more rubber all the way around the tire that it would therefore last longer. While that is true.. some manufacturers know that the added rubber that a slick has makes it heavier as well and thus remove some of that rubber from the start and the slick might actually have less tread depth than the DOT counterpart. (I know.. never easy)

So knowing what you do now.. what about the different compounds between the two choices. Well, again it remains very dynamic. I have put riders on two completely different tires in the same weekend.
To put it simple I gave him a hard DOT for an endurance race and then a sift slick for the Sprint.
I am sure you can figure out why from the explanation above. But if you would like to discuss more please feel free to contact me.
Sorry for any typos... hard to write on my phone.