One of my wrists is weaker than the other. I am working on strengthening both of my wrists, but I want to protect my weaker wrist while I workout. Should I use wrist wraps on the weak wrist or both wrists?
Disclaimer: when you are experiencing recurring pain that lasts for weeks during your workouts, then you should definitely consider getting a doctor or physical therapist to check it out. The rest of my answer is going to assume that you got the "all-clear" from your doctor and that this is in fact nothing more than a muscular imbalance in your wrists.
To me it sounds like the exercises you are doing are too intense for your weaker wrist to handle, thus the pain. I would scale back on the intensity level and gradually build up wrist strength before attempting to do these exercises at the intensity level you are at now. Additionally, you may want to consider adding some supplemental grip strength exercises to your routine to expedite the recovery. Here are some exercises that will help you build wrist strength:
As far as doing your current exercises, using a wrist wrap is totally fine, and as you scale in intensity it can often become a necessary tool in order to avoid injury. On your build up sets I would recommend forgoing the wraps, and only using them on your work-sets / exercises that are giving you problems. That being said, wraps can become a crutch that prevent grip strength from developing, and will lead to eventual degradation of wrist strength in relation to the rest of the muscles.
As for using one or both, I would personally stick with both. For one, if the weak wrist is experiencing pain from the exercise, chances are the other wrist isn't too far away either. Second, by using only one wrist wrap you will be developing the muscles asymmetrically.