TAPI is a telephony API developed by Microsoft, TSAPI is a telephony API developed by the company that eventually became Avaya (AT&T -> Lucent -> Avaya). TAPI's design was oriented torward first party call control (i.e. phones), while TSAPI was oriented torward third party call call control (i.e. PBXes and central office switches). TAPI does have some limited support for dealing with the call audio, TSAPI has none. I'd be very surprised if you could find a TAPI TSP for Avaya AES, and you won't be getting it from Avaya if you do.
If all you're interested in is call events, then you can do the same things with both APIs, but the code will be totally different. Most companies that have products that have to talk to both Avaya AES and Cisco Call Manager have created an generic CTI interface for their product and then write code modules (aka drivers) that translate the TSAPI and TAPI call events to their generic event format. In your case, depending on your OS, you can move your TAPI code into a separate DLL/shared library, then create a separate DLL/shared library that implements the same functions but using TSAPI instead of TAPI.
Frotunately, a lot of the smaller PBX manufacturers use either TAPI or TSAPI based APIs for their CTI interfaces, so if you have to connect to a third PBX, chances are good that you can reuse one of the two modules.