What does the NY Senate coup actually mean?

Okay, two democrats sided with the republicans in the NY State Senate, to declare the republicans the new majority, and appoint a republican as majority leader. Okay, fine. What does that actually mean? What changes now? Does the Majority Leader actually have rights and priveledges above that of other senators?

I always thought “majority leader” and “minority leader” were just terms used to designate the spokesperson for the party in power and the party not in power. But everyone in the Senate and government seems to be up in arms about this, as though it actually has an affect on something. Can anyone help explain?