Are there any differences between the subalkali and alkali rock occuring tectonic environments?
Yes, there are many differences and subdifferences and nuances.

I will try to keep it in simple terms which will satisfy your question.

Alkaline magmas form by low degree partial melting of the deep mantle. This is why you see them in hotspots. It's very hot down there, but you do not form a lot of melt. For example, the amounts of melts in Hawaii are negligible compared to what you have in large igneous provinces (think Deccan Traps) or mid ocean ridges. Another tectonic setting with common alkaline magmas are rift zones. The best example is the east African Rift (Ethiopia, Kenya, etc). Again - deep melting of the mantle leads to alkaline magmatism.

Subalkaline magmas form in shallower melting. For example, the voluminous mid ocean ridge basalts are not alkaline, and the melt forms within a few kilometres of the surface. Arc magmas (andesites etc) are also not too alkaline because they form by fluid-assisted melting of not too deep mantle above subduction zones.

Even though large igneous provinces form by mantle plumes, the melting so extensive that the melts are not alkaline any more. To get alkaline melts, it has to be low-degree partial melt.