Organa, Motets, and clausulae were all common in the 13th century, written and composed by the then-leading School of Notre Dame (Leonin and Perotin). However, I've found it really hard to understand the differences between all three in terms of structure and Technique, especially that samples sound very similar.Could anyone please define them and explain the differences between them?
This is a very basic answer, culled from Music History class nearly 40 years ago. Organum started out ("Early Organum") as an improvisation of a single voice over a composed voice, usually simply in parallel 4ths or 5ths. As time progressed, this became more free ("Free Organum"), with the introduction of contrary and oblique motion in the improvised voice. Eventually, the improvised voice became more ornate ("Ornate" or Melismatic Organum"), and a series of fairly standardized "riffs" came into being. These are clausulae, not unlike Indian ragas in their relationship to the music in which they occur.
The motet is a (not much) later development, taking a clausula and performing it as a separate composition, with some additions. During the Renaissance, the motet evolved to become secularized, although not exclusively so. Over time, also, the motet evolved to have a more formalized compositional structure.
This outline will give more information.
Organum was a way of embellishing plainchant by adding other voices, originally in mostly parallel motion, later in more complex counterpoint.
The melismatic portions of chants were favorite places to add elaborate organum. These are clausulae. They probably developed out of improvisational practices but some were written down and became fixed.
So your choir is going along, singing a chant in unison (this is normal chanting, not organum), but when you get to the long melisma, you slow down the chant to long tones and soloists step out to sing florid embellishments above it (they are singing organum to make a clausula). Then you continue with the chant in unison.
If you take the clausulae and add words to the upper voices, so that you have a Latin-texted tenor in long notes with faster-moved voices above it, singing different texts in Latin or French, you have a motet. These came to be separate pieces, not just ornaments to plainchant.