This is more like a demonstration record for a piano builder. Broadwood is an English builder of pianos from 1728. His quality was high enough for Beethoven and Haydn. This is a collection of works played on different keyboards from the eighteenth and nineteenth century. One of them is a harpsichord, the rest consists of fortepianos.
I will try to explain the difference in short. These are all keyboards, but they work differently. The harpsichord works by plucking the string. If you press a key on a harpsichord, a very complicated mechanism takes the string, and plucks it, like a harp. It is a harp with keys. Because of this mechanism, it was not possible to play softly (“piano”) on a harpsichord. Some models featured pedals to soften the sound, but that was a crude measure to add some dynamics.
The fortepiano was the next stage in the development of the keyboard. Here strings were hammered. Pressing a key brought the hammer to the string. Not plucking it, but simply hitting it with force. This made the keyboard a totally different instrument, and most composer didn’t write for that. The hammer could also be used to hit the string hard (forte) or soft (piano), creating endless possibilities in dynamic expression.
The fortepiano was then developed into the modern concert piano, creating a more sustained tone. Modern pianos in general have a more even tone throughout, but lack a certain charm. Fortepiano keeps being appreciated for its softer tone and gentler expression. Both have the same kind of mechanism for hitting the strings however.