From this exhibition by an astronaut on the moon dropping a hammer and a feather and showing they move at the same speed, why do the particles of the comets tail move away from the comet, slow down and fall away as it disintegrates? Shouldn't they move at the same speed of the comet? For example if the feather was attached to the hammer and it was falling and broke off wouldn't it fall at the same speed, not start to trail behind and eventually fall away?
The tail of a comet is not actually "slowing down and falling away" from the comet, like you might expect to see when smoke streams out from behind a moving object on earth. The tail of a comet is actually being pushed away from the sun by the solar winds and radiation. That's why the tail of a comment always points away from the sun, and doesn't stream out behind it.
If there were no other forces at work, the coma would actually travel at about the same speed at the comet. As the material surrounding the core of the comet vaporized, it would form a coma ("atmosphere") made up of sublimated gas mixed with dust, moving along at the same speed (more or less(1)) as the comet… like you observed in the video.
(1) In reality, without the solar winds, the coma of the comet would continue to move outward, as its expulsion velocity would easily overcome the comet's weak gravitational pull.
The tail particles from a comet are ejected, so they have a distinct velocity than that of the comet. Besides that, they orbit the Sun in an orbit similar to that of the comet when they were ejected, they don't slow down.
About the hammer and the feather, it is not the same to just break off than to be ejected. If the hammer blows the feather up while falling, the feather will delay a lot more than the hammer to fall.
There are a few more elements which you might consider. Like, when dropping a hammer along with a feather in earth, the feather descends slower than the hammer. Reason: although the force of gravity is same, their mass, and the resistance from external forces faced by them is an entirely different story. The air between the object and the ground must also be taken into consideration. Whereas, moon lacks atmosphere and hence they fall at relatively same speed (with only low gravity acting upon them).
As a comet approaches the inner Solar System, solar radiation causes the volatile materials within the comet to vaporize and stream out of the nucleus, carrying dust away with them. The streams of dust and gas thus released form a huge, extremely tenuous atmosphere around the comet called the coma, and the force exerted on the coma by the Sun's radiation pressure and solar wind cause an enormous tail to form, which points away from the Sun.
Similar to the air acting on the detached feather (as said above), the solar radiation pressure and solar winds act on the tail debris of the comet. For instance, you drop an article from a moving vehicle, it falls to the ground rather than flying in the air at the same speed. Deceleration and opposite forces like solar winds cause it to slow down.
UPDATE: The reason why mass is taken into consideration is due to the fact that mass is directly proportional to gravity. So, increasing the mass means increased gravitational pull. That is why heavier objects like hammer fall faster than lighter objects like feather. Explaining this would be a matter of physics and not astronomy which is irrelevant to this website and should be discussed in the physics stack exchange website.