With the engine together and running, a solid lifter engine will make a light clatter, as the valve lash (necessary with a solid lifter cam to allow for expansion of parts) is taken up. A hydraulic lifter engine should be nearly silent.
With the engine apart, solid lifters will fell just that (in the pushrod area)...solid...hydraulic lifters will have a little spring loaded plunger in the middle which the pushrod seats in.
Lastly, although you didn't ask, one can run hydraulic lifters on a solid lifter cam but not the reverse, as hydraulic cams don't have the clearance ramp to more gently take up lash.
If you had one of each in your hands, the mechanical would look like a barrel shaped hunk of metal with a very flat polished lower surface. The side has a concave area for oil to trap and provide lubrication for the lifter which slides like a small piston as it rides on the camshaft. The top has a depression for the pushrod to ride in and keep it centered. Properly adjusted a valve train with mechanical lifters needs about 0.020 clearance to account for growth in the pushrod and valve length as the engine heats up and parts expand. This clearance leads to a clattering sound when the engine is cold. As it warms up the clattering reduces and when correctly adjusted disappears when the engine is hot.
No body liked listening to the clatter, so engineers found a way to take up the slack and have zero clearance for the valve train. The hydraulic lifter can be thought of as two barrels, one inside the other with a spring retainer on the top to hold the central barrel trapped inside the outer barrel. The central barrel is shorter than the outer barrel and hence there is a vacant space in the inside bottom of the outer barrel. The oil area is like the mechanical lifter but has a hole in it for oil to enter the center of the outer barrel. Oil inside the outer barrel fills any vacant space and pushes the inner barrel towards the top of the lifter. This forces the inner barrel up against the pushrod and takes out any slack in the valve train when the lifter is on the low side of the cam shaft. As the cam starts to move the lifter upward, the oil holes in the block and in the lifter misalign and so the oil inside the lifter is trapped. Once oil is trapped inside, the lifter becomes "solid" consisting of the two barrels and the trapped oil. Since liquids are not compressible, the trapped oil acts like solid material inside the lifter body. It moves just like the solid lifter from that point on up the camshaft rise. Basically, its an auto adjusting lifter that uses oil under pressure to take up the slack in the valve train. The valve train is adjusted for zero lash plus an amount needed to center the inner barrel in the permissable travel range below the retaining spring. The usual adjustment is zero lash plus one half turn of the adjuster nut. The one half turn centers the inner barrel and provides for a wide range in change as the cam, pushrods and valve tops wear. For all intents and purposes, a properly adjusted hydraulic lifter will stay in adjustment for the life of the engine. Why aren't hydraulics always used? As the engine speed increases, there's less time for the lifter body to fill with oil. If not full, then the hydraulic lifter collapses and will not fully open the valve. Increasing the oil pressure can compensate but at very high rpms even increased oil pressure may not be enough. For this reason racing engines often use mechanical lifters even though they are a PITA to keep adjusted.