I have visited Cadiz for Semana Santa the last 3 years. I love this city!
The first year I stayed at the parador. Cadiz Parador
It is nice, the food is great and it is right on the ocean with a pool and great views from the balcony. The only drawback is that it is a good 15 minute walk to the center of Cadiz.
Last year I stayed at a new hotel, Hotel las Cortes de Cadiz. Las Cortes
I had seen it the year before as I was roaming around and it looked charming. I had a room that overlooked the street with a small balcony. It was a double for single use and was on the small side. The cost was similar to the parador. I would say the parador was fancier but the location couldn't be beat. By the way, I had read some reviews about the hotel and there were several negative posts regarding some rooms not having exterior windows. These rooms had a "window" that was basically above the room door. Not only was this claustrophic but they opened up to a main foyer. For this reason I specifically requested a room with an exterior window and preferably with a balcony. I saw the rooms at the hotel with the interior windows and I definately wouldn't want one of these.
More on the budget side is the Hotel Bahia. Hotel Bahia
I've never stayed there but it is in a great location just off the main square. I took a look at the lobby this year and it seemed quite nice - definately more of a budget option than the first 2 but clean and comforable looking.