Hydraulic, solid or roller lifters - they are the interface between the cam and the push rods.
Tappets came about because early hydraulic lifters tended to leak down and cause a "tap" that sounded like solid lifters out of adjustment.
The obvious problem with hydraulics is that you can't adjust out the noise.
Many, many moons ago we used the term tappets quite frequently with customers who were complaining about engine noise, especially with the higher performance cars.
"they wouldn't call them tappets if they didn't tap"
Then we would try and convince them that it the valve train is tapping just a bit at least the valves aren't burning (big problem when leaded gas disappeared).
Are we explaining automotive or motorcycling on tappets and Hydraulics.
Working from automotive, solids lifters were adjusted from the rockers on top of the head via adjustable tappet.
It wasn't until the mid 70's most automotive engines got away from adjustable hydraulics. And yes they could be made to run quietly.
On our Harley's, the adjustability has been engineered out for simplicity and assembly sake.
The only way you can get a hydraulic to run quietly is to get it depressed to the point of mid travel of the lifter.
It will pump as it should and be quiet. If it is making tappet noise, it too loose.
Just my .02