Are air purifiers or air ionizers useful against fatigue?
No matter how I’m feeling, as soon as I get into my house I start to feel tired and run down. Would an air purifier or ionizer help with this problem? And what is the difference between the two?

Maybe
If there is something in your home to which you are allergic it could be the house.
New carpet, mold, plants, smoke, pets, and new paneling can all be sources of allergic reaction.
Got to a hotel for a couple nights and see if you feel better. There are certain chains that offer allergy-free rooms with air purifiers. If you feel a lot better then you will need to figure out what is in your home that is causing the problem.
After you have located and removed the problem only then would I recommend a HEPA air purifier.
If the hotel visit does not help it may be that you are unwinding from a tough day.
Best of luck!
Air purifiers and air ionizers are Not useful against fatigue, they only help one that suffers from allergies !!!
If you decide to get an air purifier get one with a HEPA filter.
Your problem may be more psychological than physical.