Is there any difference in LED lighting vs bulb lighting or any other lighting? Someone said that the LED light doesn't have the full spectrum of light. Is that a problem?
It can definitely be a problem, especially if the LED lights in question are not white LED lights. White LED lights, at least the higher quality ones, put out enough red, green, and blue light to work fairly well for most cameras. The results might not be as ideal as a fuller spectrum light source, but you can usually get pretty close. Typical high quality white LEDs put out a stronger blue channel than red or green. Red is usually stronger than green at the wavelengths to which many cameras are most sensitive to red and green respectively. Depending on the exact colors of the filter on your camera's Bayer mask, you'll probably have to pull back blue and boost green to get the colors looking more natural.
The problem is that not all white LED lights put out enough red and/or particularly green light in the part of each of those bands to which digital cameras are most sensitive. When that is the case it becomes an issue if you want to remove color casts that were in the scene when you post process your images.
Shooting under traditional incandescent lights with a color cast is less of a problem because even when they have a strong color cast they are usually putting out at least some light throughout most of the visible spectrum. Assuming you shoot raw files, you can adjust the color response curve before converting the raw file and correct for the color cast so that things look the actual color they are.
Colored LED lights are different, though. Red LED lights put out very little light other than in a narrow band of red. Blue LED lights put out very little light other than a narrow band of blue. Green LED lights are slightly better, but they still mostly put out a narrow band of green light.
If you have all three colors of LED lights mentioned above then you can usually adjust the color response curve in post to make the light look natural. But if you only have one or two there are going to be colors that you won't be able to pull out of the raw data no matter what you do. Again, this is especially the case when green is most deficient. Remember that your camera, like your eyes, is most sensitive to green light. Half the pixel wells of a typical digital camera are filtered for green light. When there is very little light that includes green wavelengths in the scene your camera's sensor sensitivity is effectively halved!
For how this works out practically in a specific situation, please see this answer to Blown out blue/red light making photos look out of focus
The real answer is 'depends'.
White LED usually consists of phosphor (react with the chip emission) layered on a diode (such as GaN, which emits violet light 405nm) that's why they are white. The phosphor gets activated by the light from the chip and emits another wavelength of light, hence different colour in a mixture to produce white.
As you can clearly see, White LED has already covered up most of the spectrum leaving only the near 400nm wavelength uncovered. The other problem might be uneven wavelength brightness, which means even though the light looks white, the object tends to have a unnatural blue/green tint as shown in this video. The technical term for this is called low CRI.
Also, the brightness of such LED will somewhat suffer due to inherent Stokes shift problem. Manufacturers have invested to improve the GaN system such as shown below.
Then what about bulb you say?
Well, fluorescent light works with the basic principle as the LED, which means they too can have low CRI, albeit a lot brighter.
What you are comparing, I assume is incandescent halogen lighting. Halogen light have two main disadvantages, being really hot when using and very inefficient. The light they produce have great CRI (not to say the best), but are definitely one of the strongest (halogen lights on car, anyone?).
---- Edit ---- Did I mention plasma lighting? They are quite efficient and have very good CRI. This is because they mainly heat gases up (argon/neon or more) into plasma and turning them into a blackbody radiator.
Look them up.
Another difference: pulsing artifacts when used for stage lighting.
LEDs arrays are dimmed/colored by being pulsed at full power rather than run continuously at partial power.
While this is also how modern tungsten dimmers work, tungsten (and phosphor-based LEDs) tend to produce light that is an average of the power input. The RGB composition LEDs in stage lighting have spikey light output closely following the input power.
The low-end LED stage lighting I've ex[erienced cycles at around 240 hz. With means shutter speeds much above a 1/250th are likely to exhibit weird banding. Also, at slower shutter speeds you get strobing where you would've gotten smooth blur under tungsten--fast-moving drumsticks for instance.
"White" LED lights usually have an extra dose of blue in them, compared to other conventional white light sources. My own LED lighting requires post-processing to get the white balance correct, because even 'preset' white balance in-camera (calibrated using a gray card) doesn't provide enough correction to counteract the excessive blue.
Post-processing can fully correct the white balance using a two-step approach (1) high color temperature around 10000K followed by (2) blue-channel-only decrease. I have a few examples here: https://www.photoartfromscience.com/single-post/2017/09/09/How-to-Correct-an-LED-%E2%80%9CWhite%E2%80%9D-Light-Source
I'd further recommend you shoot in RAW format to get the most adjustability in post-processing.