Right now there is no restriction. Random checks do happen. But nothing should stop you from traveling between the two states. Follow covid sop and you should be good. It's always handy to have either a rapid test done or latest covid report.
Travelled to Tamil Nadu recently from Kerala. Better to take the Travel Pass online - you can apply on https://eregister.tnega.org/#/user/pass - you get the pass instantly and you can print and have a copy handy. We travelled by road and they checked the pass at the Border. Chennai - Pondy there is no checking as such that I experienced