why do cars with manual transmission slow down faster than ones with automatic transmission?

The fluid coupling between engine and transmission is almost a one way street, transferring about 90% of the power from the engine to transmission, but transferring very little the other way. The clutch locks the engine to transmission with no losses as there are in the fluid coupling, so energy is transferred either way equally. The fluid coupling works much like the ratchet mechanism in a bicycle that transfers power to the wheel when you apply pressure to the pedals, but otherwise the back wheel free-wheels and clicks allowing you to freely turn the pedals backwards while coasting along. The person above who talks about gearing is not quite correct. In a 4 or 5 speed manual transmission, the gear ratios are very close to those in a similar automatic with 4 or 5 speeds. Engine braking works in a manual only because the power coming from the wheels gets back to the engine. In an automatic, virtually no power gets to the engine from the wheels because of the slippage in the fluid coupling.

With a manual transmission you can downshift, or "engine brake" to slow the car. On top of that when the clutch is out you have a solid connection from engine to wheels, unless you are in neutral. This makes it harder for the tires to push the car, but easier for the engine to push the car.

An automatic is geared different and has a fluid transmission allowing more variability in the way you move the car. Less acceleration in most cases, sometimes worse economy, and occasionally more repairs. This and when a automatic is in gear it wants to roll no matter what, because the engine is always trying to push the car if its in gear. By the same token... an automatic is far heavier than a manual trans.

It's all about engine braking. Unless you put the gearshift selector in low gear (specific gear depends on the transmission), an automatic transmission has a one-way roller clutch in it. It lets the engine/transmission drive the wheels but the wheels can't drive the transmission/engine.

manual transmission cars are geared much lower than automatics, so they decelerate quicker and can be down shifted as autos only downshift at a complete stop when braking