A sage old piece of advice is to never split the party. It usually ends badly for the players, and some monster out there becoming extremely wealthy. “Why is that the case?” some innocent new player might ask. Is it always true? I've seen this occur a few times, mostly with horrible failure as a result but a hand full of times with amazing success. And so, I'll be recording the situations and issues I've seen in the hopes that it helps someone out there. That way it can be avoided or the issues can be minimized.
How an encounter is balanced and constructed plays a bit of a role when talking about splitting the party. If you had an adventure meant for solo play and ran your party through it, splitting up wouldn't be a problem. However, combat encounters are typically balanced around the size and level of the party. In the case that set-piece style encounters aren't used, the number and type of enemies are still roughly determined based on the strength of the players in some way. If you bring less than expected into a combat encounter, things can go very badly very quickly. It's kind of like taking some party member(s), throwing them into a room with the big bad and locking the door. The only difference is that in this case, the party members are going there willingly and locking the door behind themselves.
Of course, not everyone balances their encounters in this way. If your Dungeon Master likes to send weaker encounters that are meant to whittle down your resources, splitting the party can be a viable option. However, having everyone committed to one encounter instead usually results in a far easier time and fewer resources spent. It's just a result of typical action economy and being able to do more per turn.
Risk vs Reward
Players will make risk vs reward calculations for the positions they are in. In many campaigns, there isn't much benefit from splitting up and the risk massively increases. Most of the times this happens, it involves stealth. However, players typically attack from stealth for an advantage, which is more effective if backup is close by and more people are involved, or when trying to sneak into somewhere. The deeper they sneak in, however, the more risk.
There can of course be exceptions. If the intention is that players will split up, the Dungeon Master might try to make it lucrative to do so. However, if the threat is too high they still won't do so and prefer to go after one objective. If they know that they can go after the second objective afterwards, this further shifts the risk vs reward calculation. They also might feel it's a trick because it was never the right choice before and staying together is safer. This is one of the challenges of making it lucrative to split the party. Even if you do, they could choose not to. If you modify the encounter depending on decision of the party, it changes the nature of the choice. This is especially true if they notice it's happening after trying enough times. The choice is no longer about which choice has the best chances of success, but which approach is more advantageous from a meta-game perspective. This could be mostly a non-issue if your players don't meta-game though. Of course, the encounters should still make sense in context. How death and retreat are handles also factors in.
Ease of Running a Session
It's typically easier to run a session for one group of adventurers. This also means that this is the more common option to go for. As a result, trying to split up in such a situation leads to fighting 2 encounters undermanned. The encounters just might not have been setup for such situation. The Dungeon Master might skew things a couple of times and put fewer things to fight, but most of the time they'll just run it as it was. This does mean, however, that the adventure was intended to be tackled as a group. This is good, since it gives everyone something to do and a challenge. However, it also has a way of making splitting up not worth it.
Why is it easier? If your party is split up, you have to give both smaller parties things to do. Giving them equal attention while keeping the party members that aren't playing from being bored can be difficult. If everyone is good at what they do, the other party might have fun just watching. However, if things go unexpectedly well or badly, the attention each party gets shifted. Keeping it somewhat equal and both groups engaged is tough. It's often just easier to have everyone together. Something else needs attention? Send the NPC party there while yours handles this. If players often like to split up, what you sometimes end up with two sessions that are going on at the same time. I've been witness to a campaign where the Dungeon Master just gave each group their own time and then a week or two later they rejoined into one bigger group. I've only seen that once though.