Alhambra in Granada and/or Alcazar in Seville

Check out Trip Advisor. Someone asked that exact same question in 2009--so you aren't alone. I've seen both and loved them both. The Alhambra is more extensive because of the gardens (but in the winter that might not matter). On a budget--of time or money? If you may never go back, then go to both. If you want to save something for the next time...that makes sense too. Enjoy!
This link may work (Re: Trip Advisor): https://www.tripadvisor.com/ShowTopic-g187443-i284-k3228117-Alhambra_or_Alcazar-Seville_Province_of_Seville_Andalucia.html

On a budget time and money Thanks. I will probably save it for the next trip if I don't find out I have a rich uncle.
I got the Alhambra tickets for January 4 2019

The Alcazar is free on Mondays.

I vote to see both. They are both amazing and were the highlights of our trip to Andalusia!

free on monday is the only way I will be going there; thanks for the FYI

The Alcazar is my favorite sight in Sevilla. I thought it was different enough from the Alhambra to warrant seeing both.

Since you're there around New Year, bear in mind it is, I think, closed on both New Year's Day and Epiphany. Also, I think the free entry on Monday may only be for the last hour. If so, remember it's quite an extensive site so you would need to be discerning about where to linger as you go round. And, it's not my money of course and not fair for me to spend yours, but the standard admission is not huge compared to transport/accommodation costs, so I'd see if there is any way to afford both it and the Alhambra given you're already there. It is different to the Alhambra in many ways.

I say go to both, and skip something else, or maybe miss a dinner. Don’t skip gelato in Seville though, as there is a great gelataria very near the Alcazar (and Cathedral). I also have a great memory of a fine “chocolate con churros” breakfast at a hole-in-the-wall place near the Piaza de Alfafa, where I was invited in to photograph the whole process and the mom-and-pop operators. Not that you asked.

If you have an Alhambra ticket, and you are on a time and money budget, skip the Alcazar in Sevilla. Architecturally, it is similar to the Alhambra. Hopefully you’ll get a chance to visit it on a future trip as it is impressive in its own right.

We just visited both in September. Compared to the Alhambra, the Alcazar was a distant contender. We paid to see the royal apartment as well. There were very few furnishings, we were hurried through by the guards, and our audio guide was confusing because it described art and furnishings that were not there. Many - most - of the water features were turned off. Also, on the day we went it was very crowded and the signage was bad. We waited in the general line for those "with tickets", and eventually realized we could walk right in (in order to meet our timed entry for the apartments). I know others will disagree, but the Alhambra was amazing, especially compared to the Alcazar.
Have fun in Spain!

. Architecturally, it is similar to the Alhambra.

That is exactly why I was thinking of skipping it.