PI are definitely the better quality, but they also charge significantly more. My go-to brand is Ibis, which is slightly less plush but significantly cheaper. Note that I am talking "standard" (red) Ibis, not the budget (blue) Ibis, which is pretty much the same as TL. There is a budget Ibis at Heathrow that was all too obviously a TL previously, all they have done is taken off the TL name outside and replaced it with Ibis - everything else unchanged! I actually had to check with the receptionist that it was an Ibis. Likewise the budget PIs are nothing like the usual PIs.
I think all PI were built from new, although can have very different feels. One I stayed in had a very Crossroads Motel vibe! Whereas a lot of TLs and Ibis have been converted from something else, so there can be a lot of differences between hotels. One TL in Croydon used to be an office block, and has absolutely massive rooms. I think a lot of the Ibis hotels (and other Accor brands) are actually franchises, owned by someone else but run under the Accor brands.
TL charging for internet is surely a minor niggle - I just use my mobile data.
If you want a cooked breakfast or evening meal with Ibis or TL, you have to be sure to book a hotel with a restaurant - not all have them. I find the cooked breakfasts at TL just as good as those at PI (a cooked breakfast is a cooked breakfast), and at a standard £9.95 significantly cheaper than PI. Cooked breakfasts at Ibis equally good, but the prices are inconsistent from one hotel to another, and can be more expensive than PI - however you can usually get them cheaper if you book the night before (but they don't seem to advertise that much for some reason). Standards generally consistent across the brands, but as with everything there are outliers, and dependent on how busy they are.
All 3 chains being discussed operate dynamic pricing, with prices varying quite dramatically from day to day, and also going both up and down over time. You can therefore sometimes get a better deal in a PI - always worth checking. Also worth checking that prices haven't gone down, on one occasion the rate at an Ibis dropped from £120 to £70 a night, so I rebooked at the better rate.