What an interesting question you pose.
I can tell you, when the internet goes out for a few hours, what I miss most immediately...
Id miss being able to send instant messages to my family who lives at the other side of town, as this is our main form of communication.
I'd miss being able to check the weather (if the internet outage was weather related)... For instance a winter storm
And up until very recently where a 100% of my customer facing business was conducted on an online retail platform, I'd miss being able to stay in contact with my customers and get any work done. Offline, I can do physical work at home just fine without the internet however, while I am doing the manual labor for hours and hours (and days and days) on end, I very much do enjoy putting on Netflix and listening (not watching, I can't cuz I'm working) to favorite programs that I have honestly seen or I should say heard, dozens of times already. But that's the reason why I put them on. Because if it was something new, something that I wanted to watch, then I'd be looking over at it, and not looking at what I'm doing, which requires visual attention to detail... I do have DVDs that I could pop on, I could put on the radio, but I would miss the programs that I can stream
And if the internet really was gone forever, I think I would miss the ease of communication with certain aspects of society.
By that I mean if I'm part of a group, and that group wants to have a get together, you don't have to have the leader phone every single person to tell them about an upcoming get together, we can just send out an email.
For my day today stuff I'm very happy to do online banking, most of my bills are pre-authorized payment, but it is nice to be able to log in to your bank account and make sure that everything is up to snuff, pay off a bill if you need to that's not taking out automatically, it means I don't have to get in the car and drive to the other end of town during business hours and stand in line and wait for a teller to do this exact same thing I can do it from home whenever.
I used to really enjoy online shopping, admittedly I haven't been doing very much of it the last few years, I loved the fact that you could literally shop the world. But if there was something quite niche that I was looking for, I didn't have to spend hours and hours driving to the most obscure shops and poking in every little corner trying to hope that maybe possibly I'd find what I'm looking for. Online, you can be in the bathtub, you can be waiting for the lasagna to bake, you can be woken up early by raccoons mating beneath your window and then you can't get back to sleep so you can go online and do a little bit of online browsing or shopping...
I won't miss spam, although I have to admit I don't enjoy junk mail that's delivered to my physical mailbox either, and I wouldn't like how there would probably be more phone calls. I really hate phone calls.
I would also really miss YouTube, not that I consume much of it but I find it to be a fantastic tool if there's something I don't know how to do, more than likely somebody has posted a video which might be exactly what I'm looking for. So if there's something wrong with my car, not something major, but something like, oh shoot, how do I reset the computer after an oil change so the car realizes that I just gotten an oil change and it's not saying that it needs an oil change? If I don't know how to do that, and I can't find the manual or I can't quite understand the process, a lot of times a YouTube video is exactly what I need.
Some are way too much talking, or not enough information, or poorly presented but so many of them aren't.
I would miss being able to store my photos and videos in the cloud, yes I store photos videos and other such files on my hard drive I also use an external hard drive, I also use thumb drive backups, and yeah I still even burn data to a DVD, and if I really like the photograph then I will get a physical print out of it to put in a frame but, 99.9% of what I video or photograph ends up stored online and I love the fact that as long as you have Wi-Fi and access to your account you can pull up those photos and videos whenever you need them. (Which admittedly, most of us in the real world don't really need to pull up photos and videos of our kids or our cats doing something but we do anyway.)
From a medical point of view, the internet is great for relaying test results quickly. Like when my pets go to the vets and they get a blood test, the next day my vet can email me the results. So I don't have to make another appointment to go in, he doesn't have to phone me, I appreciate that.
There's probably a number of other things that I personally would miss if the internet disappeared forever tomorrow, but those are just off the top of my head. There's an awful lot of things that are bad and that have absolutely led to a degradation of our society that are completely internet related, the short attention spans, the obsessive need to create content leading people to do stupid or harmful or dangerous things just to gain views, the inability of being able to engage in normal social interactions... And of course, there's all the things that can lead down dark paths such as cyberbullying, pornography involving those who are not fully consenting, and the rise of absolutely whacked out groups of people who normally would never have the confidence to be so racist or misogynistic or extremist because in real life you have to find a whole lot of people to get together to support your ideas before you'll take any type of horrendous /violent action, but on the internet, you can cloak yourself in anonymity and also feel that you're part of a community and that there's more people out there who feel the same way that you do therefore you must be justified in these extremely antisocial thoughts...
So those are things that I wouldn't miss. In a lot of ways, the world prior to let's just say 1990 (just to throw a number out there) was a lot simpler, was a lot friendlier, was a bit larger, and a bit more mysterious. You had to be able to know how to read a map, you had to know how to follow instructions, you had to talk to people around you... It was just a different world before the internet had the grip that it has on most of society today.
And here we are, on an internet hosted chat group, most of us will never know who each other are or where we are in the world or what we look like or how old we are or anything else, and here we are all talking to each other and having this lovely discussion. So, that's an interesting thing about the Internet isn't it. It's great to be able to easily communicate, if you use the tool for good.