I'm not a plumber, but have played one on many MANY occasions. In real life, I'm the son of a plumber.
First thing that comes to mind, especially after your saying that the tub did drain well but is now slow, is hair/soap scum build-up. Very VERY common, & can happen in a very short period of time.
A common cause of this problem is setting the plunger-type stopper in the vent too low. It will work for a while, but since the stopper is blocking some of the line into the drain it will clog with hair & soap scum faster. If your stopper is the lever type on the face of the tub front, you should consider pulling it out, inspect it for signs it's sitting too low, then test drain the tub with it out. If the tub drains well with the stopper out, adjust it higher (go about 1/4" at a time) until the tub drains well & the stopper still works. This may, & usually does, take a few times to get just right.
Cleaned a fist-sized mess of hair/soap scum from my GF's tub drain last year & she was absolutely delighted in how fast the tub now drains (almost as delighted as she is that I'd removed the leaking tub stopper at the drain so she can use a simple rubber stopper & can now take a bath).
As for the reason for a vent... A "plug" of water moving through the drain system MUST have a way to allow air behind said "plug" otherwise the system will pull air from wherever it can, usually from the nearest P-trap, which can & will drain the trap allowing sewer gas into the house through the now dry trap.
Hope this helps!