Zostrix contains capsaicin, which probably works by reducing a pain transmitter in the nerves called substance P. The burning you describe with the Zostrix is not unusual, and usually will improve within 1 week. This has been my experience with the medication in general, though it may take slightly longer or shorter to notice the burning decrease and for the pain improvement to "kick in".
Gabapentin is usually very effective for neuropathic (ie nerve related) pain. One common problem with gabapentin is that it is underdosed--sometimes doses of 900 to 1200 mg three times per day is needed for adequate pain relief.
With regard to the cause of your neuropathy, I assume that you have had bloodwork checked for this and that you are not diabetic. If not, I would make sure that your doctors have checked at least a vitamin B12 level, thyroid hormone, and serum protein electrophoresis (SPEP) as these are treatable causes of neuropathy. Sometimes more extensive testing can be done by Neurologists if they suspect a particular disorder such as vasculitis (inflammation around the blood vessels that supply the nerves), HIV, heavy metal levels, or genetic testing for inherited neuropathy. These are not checked for every patient, however, because in most cases they are normal and testing is not warranted unless you have other symptoms/history that suggest these may be the problem.
If gabapentin and zostrix do not control your nerve pain, other medications that can be tried include amitriptyline, duloxetine (Cymbalta), pregabalin (Lyrica), carbamazepine, and lamotrigine. Each medication has its own side effects and dosing schedule, so picking the right medication needs to be a joint effort between you and your doctor.
I hope this has helped.
In my experience with severe painful neuropathy, Zostrix is helpful but does not often control the pain well enough on its own. Most of the time, I prescribe it as an adjunctive (add-on) treatment in addition to a medication such as gabapentin. For some people who find the gabapentin helps partially and do not want to take another pill, it is a good option. I would recommend sticking with the Zostrix for at least 7-10 days before judging whether it works or not.
Best wishes to you.