I was a PADI instructor who became a TDI tech instructor. I then switched over to PADI and did both briefly. Then I dropped TDI and went exclusively with PADI. My answer will be as complex as that history.
As many have noted, in its earlier days, PADI did not do tech itself--it farmed that out to DSAT, a subsidiary. I decided to become a tech instructor just a bit after PADI took over the DSAT program. I live in Colorado, where the tech market is tiny--miniscule, actually. There were no tech instructor training programs I knew of I knew of in my area, and I had never seen a PADI class or even knew a PADI tech instructor. I had gotten my tech instruction (through trimix and cave diver) through several agencies, including TDI, culminating in TDI advanced trimix in Florida. I used TDI for that because I already had some classes from them and could build from those certifications, and I happened to be staying in Florida for a while. I liked the TDI instructor I worked with there, and he had recently become an instructor trainer with his business partner. I was going to be spending two months a few miles from their shop, so that settled it. I became a TDI instructor. It had absolutely nothing to do with any comparison of agency quality.
So I went back to Colorado to teach TDI classes through Decompression Procedures in that meager market. I got a few students pretty quickly, and they did well getting to that point. But what were they to do then? There were no TDI trimix instructors around, and even though I was an experienced trimix diver, I could not become a trimix instructor because TDI required that I first had to complete a certain number of Advanced Nitrox and Decompression Procedures certifications in order to go on to teach trimix. My students were stuck with no local options. Some went to other locations and other instructors. Others just stopped.
There is more to it than I have room to explain, but I crossed over to PADI. I could become a PADI trimix instructor by having the required number of dives, demonstrating my skills to an instructor training, passing a written exam, and acting as an intern for a trimix class instead of racking up a certain number of certifications at the lower level. I was able to do all that, and I was then able to take my students all the way through the program through trimix. Since making the switch, I have had many more students than I even envisioned in that pipeline before.
I later dropped TDI in part because I saw no point in paying the fees to be in two organizations, and the PADI program was doing much more for me. There was another reason I am reluctant to talk about because it will open old wounds, but at the time I dropped, SDI/TDI committed an act that I believed was detrimental to the scuba industry as a whole and that I felt was was wholly unethical.
Since I have taught both programs, I can compare the two, and what I say will contradict the smug clichés tossed about by people who like to repeat smug clichés without investigating the facts. I find the PADI materials to be much more thorough and understandable. I find their standards to be more rigorous.