Specific encounters e.g. wandering monsters etc. are one thing; but what about the environment itself?
If they're stuck out in the wild on a night when it's 5 degrees C with a steady rain on a 30-knot south-easter and they don't have LTH, the environment itself becomes at best a noticeable inconvenience and at worst a serious threat. If they can't find a sheltered haven then even putting up tents won't be easy, and forget about a campfire for warmth. And yes, spells like
Endure Cold can be cast, but they don't last forever and they represent spell resources not available for other things. Never mind that I wouldn't want to be the poor wizard stuck trying to study a spellbook in the morning in conditions like that...if nothing else, spellbooks tend not to like getting wet...
Yup. In 5e - My players - "We cast Leomund's here in the dungeon, and we eat our goodberries. We poke our heads out in the morning."
Now, you might say "well, what about wandering monsters". Sure. That can happen, but unless I'm specifically trying to counter the party's abilities, the chances of either something powerful enough to bury the hut, or dispel it out of hand (and how does the creature know its a Tiny Hut?), etc., are going to be slim. I had an enemy seal off the crypt where the party slept in the hut with Wall of Stone. They, of course, just blasted their way out. And the encounter outside of the Crypt was still underwhelming, even though the enemy knew the party's abilities, had grunts, and was waiting for them. Sure, I could have 'killed' the party, but that would appear obvious to the party, and be considered "adversarial DMing". There is even less chance of something coming upon them in the wilderness - the Hut is essentially hidden. and not that big.
In my OSE game?
Party: Player 1 "I'll go forage for some firewood"
Player 2: I'll go with you.
Player 3: I'll set up the cooking utensils and get the fire going with what we have.
Player 4: I'll walk a perimeter.
Players 1 and 2 return with X hours of wood to burn on the fire.
Player 3: Makes a Wis check for their cooking. Success is a +1 to the rest roll. (uses up 4 rations for the meal)
Player 4: Returns from the perimeter, settles down with everyone around the fire and tells a story. Rolls a Cha check, success = another +1 to rest roll.
Players assign watches and there is a roll to see if anyone falls asleep during watch.
Players then make a Rest Roll (Con check, with modifiers) depending on time of year , weather conditions, whether they have a fire, cooking pots, and bedrolls and/or tents. Success = good night's rest (heal, able to operate at full capacity) - remember this is not 5e.
Some may called this "pixel bitching" or getting way down into the minutiae of exploration, and I'd argue 1) this is how we enjoy playing, and 2) once you've done it two or three times, it becomes super quick and fast.
Players have to manage their resources for travel (food, water, wood, if they're travelling somewhere they may not be able to find it readily, and take into account time of year (travelling/camping in the winter STINKS), and balance this with their own encumbrance, and everyone is involved, and they rotate roles so its not always (17 Cha player always tells stories around the fire, cleric always cooks the meals).
We find all of this helps build the world out, including what they encounter when they're foraging for water or wood, what they find going in and out of a place, and figuring out the best place to site their camp (comfortable? good line of sight? cave (what about its denizens?), etc.
Of course, and it goes without saying, YMMV, and many prefer to just handwave travel and exploration, unless its a combat or something intrinsic to the adventure. And that's fine. I find that for myself, the slower pace, the "management" of the adventure, and the exploration (getting there and back again), are a lot of the fun. And of course, we can also handwave things if we want to, but the game doesn't hand wave them for me.
Edit: and I agree with
@DEFCON 1 , there are likely a lot of players who came into the game in 5e who would not like this style of game. Of course, its also a style they didn't learn the game in either. My son plays 5e, and he sometimes complains about the ease of the system, and its complexity, but some of that could also be my talking about the older editions, and how they did things as a contrast.