Most teachers clear one and assume it covers everything. It does not, and the difference matters when your application is being shortlisted.
CTET is conducted by CBSE and is accepted by central government schools and by most private CBSE-affiliated schools across India. If you want to move between states later, CTET travels with you.
TS TET is conducted by the Telangana state government and is the requirement for state government school recruitment in Telangana. Many private state-board schools in Hyderabad accept it as well.
If you are early in your career and unsure where you will end up, sit CTET first. It keeps the most doors open. If your plan is government school recruitment in Telangana specifically, TS TET is the one that counts.
On your Acharyaz profile, upload the certificate for whichever you have cleared. Verified qualifications rank higher in institution searches, and schools filter on exactly this.