I've owned and used both and both are extremely well made, excellent devices.
Function :. I realize fully that this may just be me, but I found the ZAP to be easier to maintain and keep running smoothly. I had the DiD first so it also may be that I'm just a little more genesis experienced now. I got used to the draw on the ZAP really quickly, also the relatively open draw of the zap lends itself very well to a direct lung draw. If you care about that (I do!!), its awesome! One thing I like more about the DID is that the top cap screws on rather than just being held with an o-ring. I know people will say that's so you can adjust the airhole relative to the wick, but my DID does the same thing with the addition of an o-ring, so i get the adjustability and i dont have to worry about the top cap popping off in my pocket, which can,has and im sure will happen again

I needed to add an o-ring to the DID anyway because tightened all the way down the airhole is never where I want it.
Looks :
Both look amazing on my provari, although they are two pretty different looks and I'm sure the aesthetics of the final look will appeal differently to different people. The ZAP is literally made to look good on a provari, so while I would give it the nod in the looks department the DiD is no slouch and looks sexy on just about anything.
Thoughts: if I could only keep one, I'd keep the ZAP. I'm getting a lot more use out of my ZAP than I do my DiD. The other consideration would be that the ZAP is about 30% more expensive than the DID and significantly harder to get ahold of one. Having said that the ZaP's materials are in a class of their own (fused quartz and surgical grade stainless steel) I've heard that's going to change(the availability), but you can go buy a DID right now, not true of the ZAP. Which is ironic as the ZAP is made right here in the US about 800 miles from me and the DID is manufactured 5500 miles away in Greece! If made in the USA is important to you, there's one more consideration.