The difference between 576i50 (PAL equivalent, free version of the main private channels) and 720p50 (that's what German public service uses for "HD"; it's not actually 1080p, although they use a pretty high bitrate) is pretty stark, the difference between 576i and 1080p is even more obvious. Although TVs don't really have a neutral setting and try their best to mess every image up as much as they can.
All the time, and unless the images are next to each other you don't really notice unless you're looking for it. Not because there isn't a big difference, but because you just don't really care that much unless it is called to your attention.
I've found pixel-count improvements stop mattering for me somewhere around DVD quality for a lot of content. I can tell the difference between that and 1080P, but stop caring once I'm actually watching the show/film. For the occasional really beautifully-shot film or some very detailed action movies, I guess I might care a little. 1080P versus 4K, I don't notice the difference at all unless I'm deliberately looking for it. And that's even with a 55" screen at a fairly close viewing distance.
What does make a difference? Surround sound. I'd take 5.1 or up with DVD-quality picture over 4K with stereo any day, no hesitation.
I remember a clear improvement in the legibility of things like onscreen scoreboards in sports broadcasts. In the NTSC era I would squint at the TV trying to figure out if the score was 16 or 18 for a football game, and for a basketball game you just had to keep track of it in your head since the scoreboard wasn’t even persisted onscreen. Other things, like telling different players apart, are also easier these days (you can even make out their facial features in a wide shot!)
Many people wouldn't even be able to reliably tell the difference with common dpi's and viewing distances.