Trad does leave a sticky film on the glass, I don't mean sticky to the touch, but sticky to dust and so on, and by that I mean ultra fine dust!
Specialist window cleaning solutions leave far, far less than washing up liquid, but that film is still there.
Go back a day or so later (or a month later come to that) wash it with WFP and you will see a very faint sudsing effect and if you don't spend a little extra time washing and rinsing the glass will dry spotty.
Now that is an absolute fact, I know this because every week I have some shops (that I am going off to do shortly) where if I get to park the van in one spot I can use my WFP to clean them, failing that I do them trad with a pole, so I experience the above all the time.
But does the glass stay cleaner longer?
Sometimes yes, but it depends on several factors, there are a couple of panes where because of their location they are always caught by a vortex induced by the position of an adjacent building, done trad there is always a light film on them the following week, done WFP that film is not usually there...but if the conditions are bad enough then that film IS there.
I think I just come down on the side that windows cleaned WFP have at least the POTENTIAL to stay cleaner for longer, but the weather conditions between cleans is the ultimate factor in how long they stay clean.
Ian