The reason a clavichord has an aftertouch and a piano doesn't, is the design of the hammer mechanism.

In the piano the hammer is thrown, freely, over a part of its trajectory to the string; this can be seen by looking into an upright piano by opening its cabinet while depressing a key not hard enough to strike a string--the distance the hammer does not cross to the string by the time the key is fully depressed, and flush to the keybed, is the distance the hammer is thrown.

It is this innovation that separates the piano from both the clavichord and harpsichord--not incidentally permitting the piano to play piano and forte, the reason the formal name of the instrument is a pianoforte.

There is a direct, solid connection between the key and hammer in the clavichord allowing the vibrato described by owlbreath above. It is even reported, though as a piano player I find this a little difficult to accept, that J.S. Bach could actually increase the loudness of the note after striking it.

On the piano, doing things after striking the note does nothing; in fact, it is a not uncommon error termed beding, or keybeding