I haven't really found much difference when it comes down to the production process besides when it comes to using stems if available, I usually always found the process the same as producing an original: "Create what I'm thinking of."

Although I am aware that there are points where other artists want you to have specifics for your remixes, I haven't really had that, so I'm more used to throwing down an idea that I have for an already recorded / produced song and possibly using its stems in production if it's capable of fueling that idea.

I think the difference is that an original is completely open to you to be as creative as possible as an artist. For a remix, I believe it is about also being creative but also about capturing your take on the original artist's message. For me it usually is following a similar chord progression and doing a new melody. I also find myself gravitating towards strong lyrical tracks.

A very popular track is GTA - Red Lips the original and the Skrillex remix. I took the track and did a MelBounce style, a completely different take on a trap/dubstep track. I incorporated the essence of the original message by including a trap break 3/4 of the way through.