Create encounters that Wild Shape will not solve
Guards will attack random creatures. Pests such as Rats, Spiders, Cats, etc need to shoo. The guard doesn't like cats, and this particular guard has a bad habit of kicking them.
I imagine a guard would stomp out a spider. Guards like to have fun too..."Hey Jimmy, I bet you 5 silver I can slice that rat in two with one swing." "Well maybe, but I bet you 7 silver I can shoot this rat with my bow on the first shot!" Guards attack, which would force the druid out of wild shape and then...oh no, he's just a regular druid in the middle of the enemy camp? That isn't going to end well, especially on his own with no way of contacting the party.
Make the mission more than just stealth
Druids can only stay in wild shape for so long. Put on the time pressure after a combat so the Druid, who is almost out of Wild Shapes, won't be able to sneak around for 3 hours.
Thievery is difficult for small things
Spiders aren't great at stealing things. Neither are cats or rats. Even if the cat or rat is able to pick something up, no guard in his right mind would let a cat run off with the parchment detailing the evil plan to dominate the world.
Assassinations are even more difficult
Druids are crappy at assassinations. Sure, they can sneak in and sneak out, but that requires them to use both uses of Wild Shape, and they still have to kill the guy. Can the druid get some good damage on the evil General sleeping in his room? Sure, but he can't get sneak attack plus guaranteed critical hit damage like the Rogue can. The rogue could one-shot the general, but the druid is just going to make him angry, and then the guards are going to be called and then it's just a mess for the Druid.
Make use of traps
The spider crawling through this door unfortunately didn't notice that the door is trapped with a Glyph of Fireball...ouch.
The cat going down the corridor failed to notice the pressure plate in the floor and the pitfall under the rug...the cat has now taken an axe to the torso and then has been dropped down a 30' pit. I hope he has another use of Wild Shape to get out of there...
Spiders can't pick locks
What's that? The window is locked, you say? Well, guess the spider/cat/rat can't get in there. Hmm, what about crawling under the door? No, you can't do that because it's a well-made door that goes very close to the ground. Unfortunately you can't fit underneath it.
Perfect, you got inside and you are in the room with the secret plans you need to scout. Only one issue remains: it's locked inside a safe that weighs over 100 pounds. I guess your options are to drop out of wild shape and try to carry the safe out sneakily, or to try to drop out, get the safe open by force (which will certainly alert the guards in the next room) and the re-shape and book it. None of those sound like great options.
Anti-Magic and Magic Detection
Wild Shape is a form of magic, being quiet is not. Any form of simple magic detection spell that will alert the guards when magic is detected will give the Druid away, forcing them to abandon the mission and leaving them unable to return.
Sure, you can get into the enemy camp, but the instant you set foot in the headquarters a magical alarm sounds and you know the jig is up.
I imagine enemy casters know Detect Magic. No reason why the Druid-spider wouldn't instantly be noticed as he crawls his way into the room, because the enemy mage gets a blip on his magical radar and casts Lightning Bolt, Hold Person, Dominate Person, etc on the Druid.
Turns out, the enemy mage in question knows Sleep. Uh oh, the spider with 1hp was put to sleep instantly with no save...and he wakes up in a room, with 4 walls, roof and ceiling with seemingly no way out. Or, perhaps there is a stone door that the Druid can't possibly fit underneath. He is now a prisoner with no way to escape and is out of action until the party rescues him.
The Rogue has more contingencies in case he gets caught
The Rogue has a very good stealth mod, you mention. Rogues are also able to Dash as their bonus action. The Rogue was caught? No problem, they can simply run 90' each round until they're away. Enemies shooting at them? They can still Dodge and run 60' per round. They'll outpace those bows quickly.
Alternatively, once the rogue rounds the corner, he can simply re-stealth and try to continue to hide from there.
Rogues also do ridiculous damage so if somebody is about to notice the rogue but hasn't yet, the rogue can simply knock them out with non-lethal damage in a massive alpha strike, or they can kill them and hide the corpse.
Yes, Druids are very good at sneaking, but that's about all a spider/cat/rat can do. Throw something else into the mix and the Druid will find themselves in an impossible situation because they aren't a Rogue. Anti-magic also plays havoc with Wild Shape.
Another thing to consider is multi-party member missions. Design a stealth mission that requires at minimum 2 people. That way, the Rogue and Druid are forced to both take part. Maybe they have to split up at some point, maybe the door only opens if both handles are pulled at exactly the same moment, and the handles are across the room from each other.