What platform you played it on will partially influence this. Origins on console is a mess and always was; the combat is terrible, bad framerate, etc. On PC, Origins' combat works out much better. On the other hand, Inquisition's combat plays much better on console (well, controller) while the KB&M is a mess. In terms of mechanics, Inquisition is superior. Things like Focus and Guard add sorely needed depth to the battle system. Meanwhile, Inquisition has a lot less abilities than Origins, but Origins had a LOT of trash abilities that weren't worth taking. Origins was also severely unbalanced between classes, since if you weren't taking all the mages you were doing it wrong. Classes in Inquisition are better balanced, although DA2 had the best class balance of them all.
Edge: Inquisition
With combat out of the way, what about the story? The Blight has interesting lore, but Origins wastes the plotline, instead choosing to focus on subplots only tenuously connected to the overarching problem. Additionally, most of Origins' conflicts suffer from BioWare's Golden Third Option problem, as you can get past Brecilian forest, Redcliffe, and the Circle of Magi without being forced to make any hard decisions at all. The landsmeet can be similarly overcome, and the Dark Ritual ensures that players can punt consequences outside the scope of the game. Only Orzammar with the choice of King really resembles a morally grey decision with no clear best choice.
Inquisition, on the other hand, has very good focus with the main plotline. Everything you do deals with the main villain and thwarting him - either directly or through destroying his plans. Additionally, Inquisition features thematic depth that Origins lacks; the former has a strong religious thread running through it in which the benefits and drawbacks of the world's organized religion are examined thoroughly. Connected with that is the subversive way Inquisition plays with the concept of a religious symbol, and the PC's beliefs about being forced into such a role contain many, many instances of dialogue reactivity.
Finally, Inquisition is better in the way it handles C&C. The major decisions in the game have no cop-out of the central dilemma, as even an apparent one in the Orlesian plot is handled with quite a bit of skepticism and negative consequence. The War Table is also impressively reactive: you are only going to see about 1/2 of all the missions in a given playthrough based on the choices you make, there are multiple outcomes to every mission based on how you handle it, and the game doesn't spoonfeed you the correct answer.
Edge: Inquisition
Talking about the characters would take way too long, but it's worth mentioning because it's a BioWare game and characters are very important. Overall, I think both games have an excellent cast, but Origins has Shale, so it gets the slightest edge.
Edge: Origins
What else? Well, we have to talk about game structure. Now, I think the hate of Inquisition's side content is at once justified and overblown, depending on the degree. No, you can't skip all of it. Yes, you can skip most (all) of the fetching side content. No, you shouldn't have to skip a considerable amount of optional content because it sucks. What Origins did well that Inquisition didn't was finding ways to integrate side content into the main content. In Inquisition, only a few optional zones have a strong central quest that gives you a clear reason to be conquering zones. Most of the time, you'll probably be justifying the time you spend in the field by closing Rifts or restoring order in a general sense. It's all supposed to be unified by the sense of building the Inquisition, but it gets old to keep building the Inquisition by doing the same things in every zone. Origins meanwhile has fewer quests, but they feel more unique (at least, if you ignore the fetch questing, which people may forget has plagued all three DA games) and because of the strong division of the game into subplots, you constantly feel like you are doing this content WHILE also advancing the main plot. For example, Redcliffe, the Circle of Magi, and the Temple of Sacred Ashes all have various amount of side content, but they are all connected by the central thread of saving the Arl. This gives a stronger sense of purpose to the zones, which in turn makes them less boring. To put it another way: there's a big difference between running somewhere on the way to a story objective, and running somewhere to see where the map ends or to get that last icon.
Edge: Origins
Overall, I think both are excellent games, but the deciding factor to me is how much better controller combat feels and plays in Inquisition.