The red surrounding the area indicates you have an infection. The specific bacteria may be resistant to neosporin. You need to go to a physician. If the redness spreads, you require immediate attention. Do not accept an appointment for "several days from now". Ideally they would take a culture, but they will probably prescribe a broad-sprectrum antibiotic. The current protocols involving such infections are antibiotics, dry dressings, changed daily, swabbing with betadyne (or iodine) and do not get the area wet.
You should keep the area clean and dry. Do not apply any alcohol or any more cream to the burn. It will heal better left dry and uncovered. Do not pull off any scabs, they will drop off when the skin underneath has healed.
If you are concerned that it may be infected, then see your doctor.
The best way to treat a small burn is by apply lots of cold water to the skin when it first happens.
you need to get to your docs straight away it sounds to me like it has gone septic and you will need antibiotics to cure it asap