It is said that each word in the Vedas have their unique musical note (svara) and at all instances where a said word occurs, the musical note is the same; but at times, for words such as svAhA, bhavati, etc. occurring in the same section or passage (anuvAka), their respective musical note differs; why so? If Vedas have been passed on by a single lineage, why then are there differences in recitations (as an example Andhra pATham, Dravida pATham, etc.)?

The terms of Vedas have unique musical notes, and they will not change. If note does change, there will be a reason. These reasons are documented in analytical works such as “Svara Prakriyaa” and “Prati Sakhyam”. We can understand when we talk to elders.
There are no pATha bhEdam in the Vedas. The differences you alluded to is found only in the AraNyaka; there are no pATha bhEdam in Samhita and Brahmanas.
There are no pATha bhEdam from Kashmir to Kanyakumari; there could be differences in style of recitation (such as in loud and elongated manner).