The doctor likely prescribed broad-spectrum antibiotics for you and unless you're sensitive / intolerant to them, you should definitely take them as instructed. I can't see why your mother would be too worried about this. Maybe it's because she's worried about microflora imbalance (C. difficile lol) and drug resistance etc, but UTI can get really bad if it goes out of control.
Best to nip this in the bud as UTI's can spread. Take the antibiotics before the bacteria really gets raging.
Take what the doctor prescribed. A urinary-tract infection is simply the name for where the infection is in the body. It does not mean that if someone you know once had a UTI and now you have a UTI, your symptoms will be an exact match. Sometimes people who have a cold will call their doctor and demand antibiotics, which is wrong because antibiotics don't do anything for someone who has a problem caused by a virus. But your doctor took samples and heard your description of what is wrong, and he prescribed you to take an antibiotic. It is obvious he feels you have a bacterial infection, and that is what antibiotics are for. Take the whole course of antibiotics even if you start to feel better.
The only reason I would ever hesitate to take an antibiotic is if I were sensitive to them, and I would talk this over with my doctor if I knew it to be true. But antibiotics are generally very helpful, and if you have a bacteria-based infection they knock it out, and you sure don't want it to get worse!