I do often see people using them to take pics of flowers when I go to the local botanical garden, but I am curious as to what the benefit is. Just people using them who don't have steady hands for snapping pics of stationary, well lit flowers? Or is there some whole other aspect I have been missing out on in my numerous trips there?
There may be some aspects you are missing out on. I'm guessing those photographers you see may be using a higher f-stop which requires a longer exposure duration (too long for a hand-held shot).
Assuming the botanical garden example... camera is probably reasonably close to the bloom. Assuming hand-held photography, you need a shutter speed fast enough for the hand held shot. That may not be a problem depending on the lighting ... but will probably require a low f-stop. This will produce a very narrow focal ratio and you wont get much of a single bloom in focus this way.
Here's one at ISO 100, f/2.8 and it's a 1/25th sec exposure (I did use a tripod). It was on an image-stabilized lens so I maybe the tripod wasn't necessary but ... look at the shallow DoF. I didn't capture adequate DoF to get the whole bloom.
So here's another.
This time I altered the f-stop to f/11 ... that fixes the DoF problem but now the shutter speed is 6 seconds long! There isn't a living-breathing person on the planet that can hand-hold a camera for 6 seconds. For this... you NEED a tripod (and a photographer who is smart enough to get the hose-reel out of the shot... d'oh!).
The camera needs to be fairly low for this shot. But as the camera is on a tripod, I don't need to worry about back-strain and a visit to the chiropractor.
But there is one more thing. Turns out I do own a ring-flash. So here's a shot from a botanical garden (orchid competition) using the ring flash.
This is ISO 100 and I am using f/11 again... but this time the shutter speed is 1/200th sec BECAUSE there's a ring flash providing the light (and notice the extremely dark background ... there was actually plenty of light in the room but the light fall-off from the ring flash means that at 1/200th sec you don't get much of the ambient room light in the shot.
But ... at 1/200th sec. there's no tripod needed.
There are other techniques.
Suppose you've got a field of sunflowers and the wind is blowing. You want a faster shutter speed but if you open the aperture to allow more light (for a faster shutter), the depth of field narrows and you can't get the the entire field in sharp focus. So you grab a tilt-shift lens so you can dial it down to f/4 ... and while that fixes the shutter speed, you've still got the narrow DoF. But with a tilt-shift lens, you can tilt the plane of focus to match the blooms on the entire field of flowers simultaneously while *still* using a low f-stop *and* a high shutter speed. (tilt-shift lenses are completely manual, have a high learning-curve, and aren't cheap ... but they are another way to get a broad depth of field (or at least the change the plane of the DoF so everything you care about is inside the DoF).
Ring-flashes are more expensive than typical flashes and tilt-shift lenses are expensive. A less-expensive way to get the broad DoF is to use a tripod.