it greatly depends on what you want to do with the "overgrown area"
something normally should be done, once a year to ever other year, with it. just so, if a fire ever happen. the fire will not burn as hot, and will more likely burn slower through the given area.
a bush hog, is normally cheapest choice, to cut tall weeds and does it, in a fair amount of time. due to bush hogs more so 3pt bush hogs that fit on back of tractors stick out so far. they tend to be more choice for folks that have ponds / lakes. so they can partially back down a hill or perhaps little bit over the banks. to mow edges of the pond/lake. without getting stuck in the lake.
to note it it, a ""bush hog"" is actually a name of a company, but many folks refer to "rotatory cutters" as a bush hog. due to they normally only have one large heavy duty blade under them. ((well at least until ya get into the bat wing styles and 70 plus HP tractors that puts most folks way out of there range or even need))
a "fail mower" can be a safer mower to use, say you were mowing road ditches every day of the week. less chance of stuff flying out and into someones windshield. a fail mower can cut a little nicer path.
a sickle mower, think of it more like bush trimmers / hedge trimmers you might have around your house and you drag an electrical cord around or have a battery operated type. but if on a tractor. think much more heavier duty, and is more tended to help mow ditches and like. due to they can move up and down a few degrees as you drive parallel with the ditch.
your ""finishing mowers"" are your "5hp push mowers" riding lawn mowers, normally belly mowers on tractors. these are more like your every week mowing type of mowers. that can give like golf course like grass pending on how much of green thumb you are with your lawn.
there are many other types of "mowers" or way to deal with things.
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i thick / tall weeds and grass. it can be very easy to run over a baby dear, a baby cow there mothers put them down in tall stuff to hide them from predators, and the babies think the tractor is a predator, so they stay perfectly still, even after it is way to late and they get hit by the blades
also kids toys, to many other things can quickly be found and you may not even notice you hit them till after it is way to late.
if you have a FEL (front end loader) with say a general duty bucket on it. i tend to keep it just barely off the ground say 2 inches or as ground allows me. when mowing in woods / pasture. so if a tree limb is down or a stump is there or a large rock. i am more likely to hit it with the bucket and more likely to notice it and possibly move it up and out the way. vs letting it find it way back in the blade.
bush hogs / rottory cutters. can be heavy on a back of a tractor. more so than other implements weight same amount. due to how far they stick out the back of the tractor. and can cause quicker "teter totoring" of the tractor on the back wheels causing front wheels to come off the ground. if you have a "FEL" front end loader it can help balance out the weight. but if you do not. then would suggest extra weight on the front more so if you have any hills you need to drive across or deal with.
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there are different ways to set the bush hog / rotatory cutters. if you set the front of them so they are lower to ground vs rear of them. weeds / grass only gets cuts once, and you end up with long stalks like if you used a weed-eater. and can leave large piles of dead grass on top of everything. to me this is no real big deal in woods or pasture.
if the rear of the bush hog / rotatory cutter. is lower than the front end. you can kinda mulch the tall grass / weeds. due to everything can get cut up a couple times before leaving the bottom side of the deck.
to note it though. if you to have much difference between front and rear or vice vs. you can create low spots and high spots.
again the pile of dead grass / weeds left behind from just a quick cut is fine for me. generally i am just wanting to nock down the weeds. to within a couple inches of the ground. this keeps me from having to say "start a fire and burn things off" and can keep pretty descent green area. granted the grass is taller than say front yard is. but it stays lush and green. and more likely does a much better control of keeping washouts from happening vs letting things just grow and grow for ever. also by cutting it a couple times a year. i can ensure no major thorn bushes or thorn trees take root. and if they do, a quick cut of the mower normally nocks them down quick and easily. and if i keep up on the maintenance of keeping things half way mowed down say 2 foot or less. i can keep things in control.
again to me the areas i use a bush hog / rotatory cutter. is ""not lawn"" but pasture / wooded areas. or places were i do not need to walk all that often nor need to see down to the dirt within an inch. it is rough cut. rough looking. and used primarly to keep major weeds like stuff with thorns on them down to very min. if the old cut grass kills some stuff out some, no big deal to me. it is rough cut anyhow.
if you want a finish yard look like most city folks yards. then a bush hog will cut everything down to a point were you can then start using a finishing mower. and keep the dead cut grass down to a min.