Say I have a monk who took Skill Expert. They have proficiency in both stealth and perception. The monk wants to apply the expertise to either of these two skills. Both are essential for the monk, so which one do I choose?

In terms of general utility, Perception is better. Stealth tends to be more niche, and as a Monk you can't try for Hide as a bonus action like a Rogue. Ultimately, though, it's your call, and there's not truly and "right" pick.

Which skill do you find yourself wishing that you had a higher bonus in more often?

It depends on the game. Stealth is useful if there's stuff to hide from and the DM actually lets you hide; Perception is useful if there's hidden stuff to notice. There's also the "is someone else good at this" factor; scouting is frequently done solo, and you can't notice a given hidden thing twice. Perception is probably more useful on average, but not by enough to outweigh campaign-specific issues.

I would probably choose perception, as a monk is not supposed to be played stealthily (in most situations anyway). This would probably be best for your character later on. If you plan to multiclass into rogue later on, then I would probably choose stealth. But because you specified monk, then I would choose perception, because that is how a monk is usually supposed to be played.

That’s entirely dependent on the campaign and who else is in the party.

Sneaking without being able to notice things isn't particularly useful. You need them both to be pretty good, and both DEX and WIS affect your armor class, but only DEX affects your to-hit modifier, so your DEX ought to be higher than your WIS, usually. Thus your Stealth will naturally be better than your Perception, thus you should get Expertise in Perception to even them out somewhat.

Quote from IamSposta >>

That’s entirely dependent on the campaign and who else is in the party.

 Exactly true. In the campaign I’m in Perception doesn’t come up too often. There tend to be more opportunities for stealth. But another campaign could be the opposite. And if there is already a stealth character in the group you might not want to step on their toes, unless they want the company. 

So take what you think you will get the most out of your campaign. 

As Sposta says, it really depends on the party. I tend to find neither would be a game changer.

Stealth is good for ambushes and sneaking, but you're often heavily dependent on the rest of the party either being stealthy or at least cooperative. Perception tends to be so popular that it only becomes important for your character to have it to spot bushes.

I'd lean for perception because even if others have it, you can be the top lookout for the group because you have expertise (they generally don't). Stealth tends to be more niche - but if you have a party if ninjas, it could really help.

To echo, it definitely does depend on the party makeup and campaign. I'd probably go perception just because I know my DM uses passive perception a ton, but I've also played in a campaign where our monk was the only sneaky one and we relied on him for dungeon delving.

If this question is not just academic, I'd also recommend asking the DM which skill might be more useful in the game long-term and take that into consideration.

Sometimes when I need to make a decision like this, I think about who my character is. This feat represents an intense period of training or an impressive natural talent. Which fits my concept better? Which makes more sense in the context of my backstory? Which would I prefer to incorporate into my roleplay? If you haven't already, use this choice to explore who you want your character to be in the world.

Both options have great mechanical benefits, sure. And a balanced party is a good thing to plan out. But you'll survive with either choice. I'd let your character make the decision.

Unless your character is built around Stealth, Perception is usually a more useful skill in general, with many built-in mechanics involving it such as search, surprise, hiding etc as well as usually being the skill check most often asked for by DM. 

While Dexterity is king of ability, Perception is king of skill.