Why some of AACSB accredited universities online Master degree courses require written examination while other does not?

It is just the university, each has their own requirements for graduation, number of hours, required classes and so on.

Some have or offer both, a thesis program and a non thesis program.

Many online universities require written examinations which are proctored at a site to ensure test validity. Others have online testing with a proctor. I'm seeing this more than ever. As a test proctor, I'm checking identification, ensuring no devices are present, etc.