A 1080p screen has 6 times as many pixels as an NTSC DVD.
A 4k screen has 24 times as many pixels as an NTSC DVD.
DVDs are not well-encoded, and the sources are typically poorer, too.
DVDs store MPEG-2 Part 2 (H.262) video streams. It's an extremely old, inefficient codec. (It was published in 1996! Next month, it'll be 30 years old!) It looks best when the encoder is given a bitrate limit north of 20 megabits per second, but DVD-Video has a hardware limit of 10 Mbps, and that includes the audio and subtitle streams. Most video streams on DVDs get 4-5 Mbps. MPEG-2 also isn't a very good codec; no matter how much bandwidth you get it, it's never really considered to be “transparent” (that is, encoding artifacts are always visible).
If you take a Blu-ray copy of a film (FHD or UHD, doesn't really matter), scale it down to SD resolution, and run it through a good HEVC (H.265) encoder, you'll usually find that a DVD-equivalent encoding takes about a third, maybe a quarter of the space. Or, if you go the other way and let the encode take as much space as the MPEG-2 one on the DVD, you'll almost certainly see an obvious difference, particularly in action scenes.
Starting a physical media collection? Fantastic. Good for you (seriously). But get Blu-rays wherever possible. You'll mostly have to forego the thrift shop, fine, but if you're ever actually going to watch the film, you'll vastly prefer it.
I recently purchased the Firefly Blu-ray and it was an interesting case because it's image quality isn't that much better than the DVD (but definitely better) however it's sound quality was astonishingly better than the DVD. I imagine this has a lot to do with the source material, how it was mastered, etc. I still stream, but I like that I have a core collection that will never disappear without warning, or be edited behind my back (which happens all the time, without notice, especially on YouTube and on Amazon Prime).
When using subtitles, another reason is the higher-resolution fonts.
While the resolution may be higher on streaming, the bitrate is often significantly worse. Beyond that Netflix has done upscaling in the past with middling success.
Nevermind the horrendous AI upscaling they tried last year. https://futurism.com/netflix-ai-upscaling-old-shows-horrific
Or hooked their DVD player to the HDTV with an RCA cable and were disappointed.
On the flipside, if you had a DVD player capable of progressive scan and Component or HDMI-out, it's fine for couch viewing.
That said, there are plenty of DVDs out there (extreme case, single layer DVD with extras on same disc as movie) where the bitrate can show, but that's not a fault of the format.
If your TV is under 50", I don't think you'd notice quite so much.
Would a bluray look better? Probably. Does streaming look better? Often, but not always; streaming likes to measure quality by resolution, but more pixels doesn't help quality unless it comes with more bitrate and/or improved codecs and/or improved encoders... And sometimes streaming has pretty low bitrate.
I have been watching a number of french and mexican movies from the 50's and 60's these last few weeks and video resolution was not an issue. Sound quality and mixing on the other hand was more of a problem if I didn't wanted to turn the volume too high, especially the mexican ones (Cantinflas).
I don't know what is it with mexican movies, even movies to this day tend to have a terrible sound mixing. It is annoying because actors tend to speak in a much more natural and pleasant way than their US counterparts and their ugly vocal fry (women) or ridiculous mumbling (men).