Any visit to Cadiz should go hand in hand with a visit to Jerez, and vice versa. Jerez, home of sherry, and Cadiz, the land of fried fish, are musts for any foodie (or, erm, drinkie - why isn't there a word for that yet?). Even if you're just on a Cadiz shore excursion, you have time to get to Jerez. I picked Cadiz and Jerez for my article on Two-City Weekend Breaks in Spain.
See also:
The Best Way to Get from Jerez to Cadiz
Buses and trains between Jerez and Cadiz depart and arrive at the same points, cost around the same (five to ten euros) and take roughly the same amount of time (around an hour), so there's no real difference. As the train is more comfortable, I'd go with that.
Traveling by Train Between Jerez and Cadiz
There are both mainline and Cercania (local) trains between Jerez and Cadiz. The Cercanias train is a little cheaper but a little slower, but there is little difference: I'd just take whichever is available at the time you want to travel (just make sure you use the right ticket machine). Use this link to book train tickets in Spain or this one to check Cercanias train times (book in person for these).
Traveling by Bus
The bus station moved to in front of the train station in recent years, but the temporary nature of the building suggests it might move again, so check with your hotel about the whereabouts of the bus station (the map I was given at the tourist office was actually out of date).
Buses between Jerez and Cadiz are run by TG Comes.