None, in this case. Some techs give you overall ability bonuses though, so they can be worthwhile getting if you skipped them previously.

Hello. I was the one trying to trade for the lower tech. My reason was for tech stealing during invasions.

I hadn't done any research on the Phasor line. I had stolen Phasors, Phasors III, Phasors IV and Phasors V from invading planets. I wanted to trade for Phasors II, so that on the next turn when I invaded more planets I didn't steal a (to me) worthless lower level tech, I wanted to keep stealing my enemies highest level techs.

The AI wouldn't trade, and of course, in the next turn when I invaded worlds, I stole Phasors II, which was what I was hoping to avoid.

The AI I was trying to trade with was not the one I was invading. Both civs just had research done on that line.

Oh, I was trying to improve the AIs tech. It was fighting a war I wanted it to win, and its ships were underpowered. Would have been a great way to kill two birds if it had accepted the trade.

The question in the original post was why would I (the player) want lower level techs.