There are many aspects, differences and reasons for grounding.
I will start off with reasons why you should or might need to ground an antenna.
-your antenna requires an RF ground to function correctly.
The antenna may not have a ground plane or counterpoise to effectively match the antenna. A counterpoise can be a wire or radials usually 1/4 wavelength or longer than the frequency of intended operation. Some ground planes will work independently of a physical ground. It is still technically an RF ground, just not earth grounded. For example my CLR-2 5/8 wave cb antenna will work happily regardless of whether I have it earth grounded or not. Some ground planes or RF grounds employ ground rods berried at the base of a tower or antenna mast using the earth as an RF ground counterpoise for the antenna to work against. Some installations use radials stretching outwards from the base of a tower or mast like spokes on a wheel. This creates a ground plane for the antenna to work against, the more radials the higher the efficiency. Some antennas using matching transformers, such as baluns or ununs require a ground to function properly. A common scenario is a wire dipole. It can be fed with 450 ohm twin lead, or open wire, and work efficiently when connected to a suitable transmitter, however this means the feed line also radiates as part of the antenna bringing RF into the shack. Perhaps you want to connect a dipole such as a G5RV to a transmitter with a 50ohm unbalanced output fed with coaxial cable, the antenna will use ladder line or twin lead to connect to a balun (balanced to un-ballanced) transformer. If the balun is not grounded, the coax will radiate RF on its shield. Some long RF grounds might require being impedance matched, there is a simple solution using a chunk of coax as a grounding conductor to bleed off RF to ground that the standard copper wire may not be able to do. RF acts very differently than DC or low frequency AC on conductors, and sometimes what seems a good DC ground may choke RF and not allow it to flow to ground.
-lightning protection and safety ground.
Lighting protection is one of the main reasons to ground an antenna if the antenna does not require a ground to function properly. Such grounding will help protect your system from near strikes, but if you take a direct hit, you will usually suffer some damage. The amount of damage depends on the grounding system. Most hobbyists cannot afford a grounding system capable of dissipating a direct hit with lightning such as the systems employed by commercial broadcast antennas. These grounding systems cost thousands of dollars and are usually out of the budget for most of us. Spend as much as you can afford on your grounding, largest conductors you can afford etc. Lighting grounds must also be as short and straight as possible, smooth (such as solid copper or copper pipe, avoid braided or twisted conductors if possible, though 99% of the time that is all that is available, anything is better than nothing) So you've grounded your antennas with ground rods, now is a good idea for a safety grounding bond. This may not only be required by local codes, but is a good idea. This is a bonding conductor that connects between the electrical service ground and your antenna grounds, bringing them to the same voltage potential. My grounding system was installed a bit at a time due to finances. At one point my antennas were grounded from their feed points outdoors, but not bonded to the electrical service ground. I found out first hand what the purpose is for having a safety bond. I was adjusting my SX-99 receiver, which has a grounded power cable installed, thus grounded to my electrical service, I reached over with my other hand to power up another receiver that only has a 2 prong plug, so its ground was provided by the antenna coax to the antenna ground rods. I received an unexpected jolt. Not enough to be lethal, but enough to surprise me. Both grounds were near 30v different potential.
-Entrance grounding and equipment grounding,
If you have alot of interconnected equipment like myself and others, you may want to employ an equipment ground. This is a ground line run into the shack feeding a bus which you ground all of your equipment to. This reduces RF on the equipment, noise, and keeps the chassis all at the same potential to prevent more extreme damage should you take a lightning strike and have the discharge enter the shack. This is not a fail safe ground, but it will reduce the chance of the discharge entering one radio and traveling through it and possibly many others looking for its way to ground. Ground all of your equipment to a common point to reduce the possibility of ground loops, which can be a nuisance and a pain to diagnose if you have a handful of interconnected equipment.
Entrance grounding is the final stand at keeping lightning from getting into you shack/house. It usually employs grounded couplers or lightning restorers (more appropriately called static arresters as lightning will surely destroy them and get through) mounted on a large ground bus and bonded to your EXTERIOR grounding conductors. This is usually installed just before the feedlines enter the building.
My current grounding system that has been a work in progress for a few years consists of 4 ground rods grounding my antennas at their feed points either via the metal brackets bonded to the coax connections, or to the appropriate ground terminals on the balun/ununs. The rods are all bonded with #3 awg copper wire, and the antenna ground system is bonded to the electrical service ground with #3 copper. A length of #3 copper runs into the shack connecting to a master bus for my equipment ground consisting of a heavy bus with many lugs attached. from there smaller conductors ground ALL of my radios and equipment, even the audio processing, PCs, and backup power supplies using provided ground terminals, and if not available to a screw that makes good electrical contact with the chassis. A #4 braided conductor goes from the main bus to a second bus where my antenna switches and low pass filter are mounted providing additional grounding. I did notice that when the equipment ground was installed along with the safety bond, my noise floor dropped 2 S units. So at the very least I made it easier to receive signals.
To sum it up, yes ground, spend as much as you can afford, and don't be afraid to add on to the system to make it more reliable later on. Will it save your equipment from a direct lightning strike? Probably not, but it will increase the survival chances. Make sure your grounding complies with local codes, and your insurance covers lighting damage, that way if something is to happen, you will be covered.
all in all, any ground it better than no ground.