Among the many innovations in science, math, and the arts during the era we are discussing was the invention of modern tonality. To the modern ear, music from the periods we've discussed so far can seem rather aimless or random; by the Renaissance, harmony was somewhat better understood and the music sounds more like music. However, a coherent system of tonality was not developed until the 17th and 18th centuries.
Tonality is based around thirds, which are musical intervals--an interval is just the relationship between two frequencies; if you take two taut strings and set one vibrating in two waves and one vibrating in one, that is a simple interval (namely an octave). A third interval is what you hear if you play two keys on a piano separated by one note; in terms of frequencies it is a 6:5 ratio (minor third, used in Western music to suggest a sad or contemplative mood) or a 5:4 ratio (a major third). The traditional definition of tonality is a centering of music around a triad, which consists of a minor third on top of a major third (a major triad--for example, C, E, and G form a C major triad) or a major on top of a minor (a minor triad, like A, C, and E for A minor). There are 24 possible triads, and each is associated with a key. The fact that fifths and thirds were consonant intervals had been known since at least the 10th century (our motet begins and ends on a major triad, although Thomas Morley had little conceptual understanding of tonality), but the systemization of them like this was a development of the 17th century that made possible much more formal, complex, and systemized music. The key difference between tonal music and Renaissance music is that the keys were not just understood as a single, consonant chord, but were extended to their relationship with a scale--that is, a C major triad is associated with the scale C, D, E, F, G, A, B, C, giving a full set of scale "degrees".
This understanding led to all sorts of new flexibility in music--for example, now composers could take a chord--say, C major--and shift it, not chromatically by moving each note one semitone downwards to a B major triad, which sounds like a tonal contrast, but rather tonally by moving each note down along the scale, to what is known as a B diminished triad and sounds more like a tonal development. This understanding of the relationship between scales, keys, and triads is the foundation of classical music theory and composition, and is the reason why by the 17th century music sounds familiar to the modern ear.