jgayman has it, and one other difference is blade speed; a bush hog's blades typically rotate more slowly than a mower. Their cutting ability is dependent more on inertia of the much heavier blades than on a lighter, sharper blade spinning faster. A mower deck does have a small shock absorbing ability, through the deck belt, either in slippage of the belt on the sheave or by the belt tensioner. A bush hog has no belt between the drive source and the blade, so the blades being free to 'give' or bounce out of the way is all it has to cushion driveline shock, unless you have a slip clutch or shear pin, either is highly recommended.
I've used my mower deck often for clearing light brush, nothing bigger than my ring finger. You do have to be more careful that you don't run over a hidden stump and damage a blade or the mower's transmission. When cutting brush, I always keep my deck as high as it will go.
I've chipped up enough small dia. limbs in the way...but most of the time they were in the way because they were rotted enough to fall off of the branch nearby to the edge of the lawn vs a ring finger sized green.
Neither type slow the deck down and pretty much turn into a mulchable sized debris trail. Anything still in the way on the next cut...is pretty much chewed down.
A few years ago I chose to pull out my 30 gallon pull behind sprayer while doing the early Summer hosing of the Round Up. I'm a corner lot with some acreage so I used a bit on the curb line where the concrete and asphalt mate up but leave enough of a crack for weeds to settle in...about 1500 feet of curb, driveway joints, around the foundation....as well as hosing down poison ivy and kudzu starting to creep. I had about 5 gallons of the concentrated strength...so I hosed down around the brush line to stop anything from progressing forward or up.
As soon as I finished that perimeter of about 500ft....I get whacked on the back of the head by a POed wife who ran down to stop me when she saw what I was doing. Yelling and making a scene...with my ear plugs in, I didn't hear a thing...which only made her more mad. I'm not sure why she cherished those bushes and brush but I got the impression NOT to herbicide that area again.
I could understand if I went in and hosed down her garden....but the brush line? Anyhow...fast forward to recent time, what grew back I have to trim down. NOT for shape mind you...only to protect my arm, bare knees and face from trying to do my job in cutting the grass up to the brush line. I'm sure we've all bent down to get around a lowering branch only to sit back up and get that branch hidden behind the lowering branch smack in the face or chest. I know every sq.ft. of the lot....but some of these things sneak up on you between seasons.
I remember fussing through cedar branches that stretched out beyond the bed...with my head down so I didn't catch one of those spears in an eye. By the time I lifted my head back up....WHACK...right into a limb that wasn't going to move....so it made ME move by knocking me off the mower. The seat switch did what it's supposed to do and stopped the blades but I still had to get up and chase down that runaway Deere.<G
Be that as it may....I'm thinking the mower can handle the junk brush, twigs and small dia. shoots. If the hedge trimmers won't cut it....the John Deere doesn't need to deal with it. Saw it off and toss it in the middle of the brush and move on. The 1988-89 JD 180 is built like a tank...it just has the age on it that gives me a little concern....but if recip. hedge trimmers can cut it down...the 180 can chew it up.