I would think that recording Black Metal, the worse gear the better.
Bad gear sounds bad. Certain stylings of black metal were developed based on what sounded cool with high distortion and low fidelity tape recordings, but simply using shitty gear is going to sound like shit. Some sounds came from musicians finding a way to create a sound that fit their music with limited resources, but the quality of that music ultimately came from the creativity and vision of the artist. For the same reason, there are plenty of shitty bedroom black metal bands who try to copy those techniques and make worthless music.
If you are familiar with Burzum, he used a tiny Peavey amp and a headset to record his fourth album.
Varg used a Peavey Special 212 on the first three Burzum albums - exclusively on the second and third, while the first one had one guitar through a Marshall and one through the Peavey. While it is inexpensive, is not a tube amp, and is pretty much the antithesis of the warm tube amp sound that many guitarists lusted after, it's one of the greatest guitar amps ever. Cold, crunchy, and high distortion. It's nearly impossible to fuck up putting a microphone in front of it, the speakers are pretty much perfect for recording, and the sound is very easy to mix. Both the Special 212 (Burzum) and Bandit 112 (Emperor, others) were utilized by Pytten - an experienced and savvy recording engineer - on early 90s Norwegian black metal recordings. However, Pytten's engineering was phenomenal and a huge part of that was how he was able to match the tools he had to shape a sound that matched the music of the bands he was recording, thus why Burzum, Emperor, and Gorgoroth all sounded very different on albums recorded around the same time.
The fourth Burzum album created an aesthetic with the distorted, clipped headset vocals and the guitar tone being a fuzz pedal run through a home stereo amplifier. That aesthetic has become iconic in conjunction with the music. However, the aesthetic has been emulated to astounding failure by thousands of shitty black metal bands.
Back to the main point
We did use the Zoom to record our material live. It came out okay, but it still doesn't have the sound like these records I listen to.
What is the difference in using a microphone like the Shure vs the Zoom H4N recorder? I feel that the Zoom does a good job with the drums as everything is quite clear. What about our vocalist's microphone instead of the Zoom or Shure? What is the difference there?
The Zoom has two microphones which are relatively small and configured for live stereo recording. That may be alright for drums (I've never tried that) but it's not desirable for recording anything out of a speaker. A microphone like the SM57 is ideally responsive and sensitive for recording an amplified instrument out of a speaker, and the frequency response is desirable for guitars/drums/vocals. One of the primary benefits of a microphone like the SM57 on a single instrument is that picks up the primary source of sound directly in front of it well and doesn't pick up background noise - that's different with something like the Zoom that's meant to record a whole room's sound.
You can use your vocalist's microphone - results will vary depending on specifics - to record guitars or drums. Vocal mics generally have a pop filter (that foam thing with the cage over it) and a slightly brighter mid-range, but in the case of the Shure SM57 vs SM58, the SM58 works pretty damn well for anything the SM57 works for. If you use an SM57 for vocals, you'll want some sort of pop/spit filter.
The most important thing is to learn how use the tools you need, what the tools do, and how everything responds to certain adjustments. You don't need an expensive set of tools, or even a complete set of tools - you can take every nut off with an adjustable wrench, but there's a reason people prefer a full socket set if you have them. You can plug your vocalist's mic into the Zoom that you have (I just looked this up.) Experiment with mic placement. Experiment with mixing. If you don't have a microphone stand, poke a hole in a cardboard box and stick the mic through there. More recording gear doesn't help if you don't know how to use it, but once you learn, you flexibility is somewhat limited by your equipment. Much like instruments, you can do a lot with a moderate amount of equipment, but there's a big difference between $100 and $500. Learn how to work with what you have and read everything you can find, because most flexibility is afforded by putting together a theoretical understanding of how things work and react to different setups.