There aren't that many classic players famous for their tone that used onboard guitar buffers. Jerry Garcia is the biggest one I can think of. So if you are looking to emulate classic rock tones with classic gear, you can probably stop reading here. There are however a number of very good, scientifically sound reasons you may want to buffer your guitar's signal. Mitigating high frequency tone loss from guitar cables is one. Take your guitar's cable - just a normal 1/4" guitar cable. This has two electrically conductive pieces (wire) separated by insulation (a dielectric) - that is more or less precisely what a capacitor is. In this case this "capacitor" is, if you had to draw a circuit, across the positive and negative outputs of your guitar, like another tone circuit bleeding out high frequencies. The longer the cable, the higher the capacitance and the greater the high frequency loss. This is why some players prefer shorter cable runs, or spring for very expensive low capacitance cables - both of these things reduce high-frequency tone loss.
As a side note some players prefer longer runs as it smooths out the sound of their guitars. It is unclear if this is purely due to what was available at the time or sound preference, but as an example Jimi Hendrix used a coiled guitar cable. Personally, even if you like this tone I'd recommend just using a nicer cable and rolling off your tone knob a tad - you can undo it on the fly and it does the same thing.
As a lesser benefit, there is also the damping factor of the guitar vs. what it is driving, here either the input of your amp or some effects and then your amp. Damping factor is defined by the impedance of what you are driving (here the input of the amp) divided by the output impedance of the driver (here the guitar). The higher the damping factor, the greater the control the driver has over what is being driven. The easiest example to think about is probably an amplifier driving a speaker - a hifi solid state amp driving an 8 ohm speaker can have a damping factor in the thousands - even at very low frequencies it has a lot of control over the driver and what it is doing. A tube amp may have a damping factor of 20 say - there will likely be more variation in what the speaker does vs. what is on the recording.
So this brings us back to our guitar example. It may be hard to think about the guitar "driving" anything but that's what it's doing with the input of your amp. Passive guitars also usually have very high output impedances - so this leads to a lower damping factor and can add to the (sometimes desirable) squishy guitar sound - the guitar is actually having trouble "driving" the load.
Unity buffer vs. booster: Ok so you might be thinking that just because you have a battery-powered device in your guitar, it must be boosting your signal, right? Well no in fact, not unless it is designed that way. If you do not want a louder guitar signal (might be called a "clean boost" or something like this) you want what is called a "unity buffer". Essentially what this does is take exactly the voltage signal it sees and repeat the exact same voltage/signal, just having been "buffered" with a circuit that can drive a much tougher load than your guitar and which has a much lower output impedance. There are some onboard booster products, which should provide the same benefits described here as well as a hotter output - that's more up to you and what you want to change in your guitar, but here I'm talking about unity buffers.
I've installed and used a unity buffer on one guitar so far - in short I was stunned at what a difference it made in the sound of my guitar. It took a muddy dull guitar and made it sound "hifi" for lack of a better way of saying it. It definitely brought back a good amount of high frequency response, and it also improved the amount and quality of bass. Playing into a Mesa Mark V on channel 3 with the typical "smiley" EQ (boosted bass and highs) I was able to get a very solid metal tone, and on the clean channel the sound was very well defined.
I need to test this myself, but depending on where the buffer is I've read that you won't get a tonal change when you use the guitar's volume knob like you would on a normal passive guitar. Anecdotally that feels true to me from my personal experience, but I want to actually simulate it to confirm it is true.
So why not do this? Well a few reasons - I think as I mentioned above if you are going for most classic sounds they weren't made in this way - I'm not saying you cannot achieve them, it may just require more tweaking/EQing/rolling off the tone knob. Also it adds expense and weight to your guitar, and requires the use of a battery (usually onboard). It is also possible that your volume pedal, wah pedal, and fuzz pedals (these are the most reactive with your guitar electronics) may not sound right or work they way they used to work for you. Lastly, if you are already playing through a wireless system you are effectively buffered already, so you may not see any functional benefits.
Personally, I'm going to keep this as an "ace up my sleeve" for builds where I don't originally achieve the tone I had in mind (especially where the tone is too dark), and as a recommendation for people who play frequently with a mix of shorter/longer cables and want the same tone every time.