EDIT: LOL, in the time it took me to write this up y’all have answered it much better! I’ll keep it here in case saying it a different way helps anyone, but yeah. “What everyone else said.” 
The thing is, there is no 100% correct. Sometimes you just have to understand what’s going on under the hood to make the right decision based on circumstance.
I think understanding this will answer most of these questions for you.
Let’s start with plain-old-unbalanced audio. It’s voltage going over a wire. For electrical reasons that you don’t technically need to understand, voltage can’t happen without a lead and a ground. So to send voltage over “a cable”, two wires are actually required: one for the lead (sometimes called “hot”) and one for ground (sometimes called “earth”).
This is what a TS cable is. It contains two wires, one of which is tied directly to the tip, and the other of which is tied to the sleeve. When plugged into a device, the part of the device that touches the sleeve uses it as a ground and the part that touches the tip uses it as a lead. Cool!
But (again for electrical reasons you can just take my word for) voltage traveling over a wire creates fields that can extend a surprising distance around the wire. And anything that interacts with those fields can effect the voltage going over the wire. This can cause noise (or, if not “noise” per se, signal that you didn’t intend to travel over the wire).
Balanced audio prevents this in a kind of ingenious way. Rather than just having two wires in a cable (one for the ground and one for lead) it has three! One for one for ground, one for lead, and one for the inverse of the lead.
Why is that helpful? If anything affects the fields of the wires, it will affect both leads the same way. Then, on the other end of the cable, the device inverts one of the leads and sums it with the other. The parts that are signal are the same and just get louder. Cool! But the parts that are noise are now the exact opposite of one another and cancel out! Genius!
But for this to work, your cable has to contain three wires connected to three different parts of a jack. There are two common ways to do this: TRS, which adds another segment to the jack called the “ring” that carries the inverted signal, and XLR, that provides three pins, one for ground, one for lead, and one for inverted lead. (This is a key point: XLR and TRS are just two different jacks for doing the same thing: Connecting to a cable with three wires)
This is, as I say, really cool. But you don’t really need to worry about cancelling out noise in your cables if there’s no noise there to begin with. Cables are all wrapped in insulating rubber, so for short solitary runs, there may not be any noise, so (as others have pointed out) you may not need a balanced cable. An unbalanced cable (having 1/3 less wires) is usually cheaper. Can you just use that?
Usually, yes. If you look at a TRS connector, the ring (that carries the inverse signal) is where the “sleeve” of a TS connector is located. That means, if you plug an unbalanced TS cable into a balanced port, the inverted signal will be sent to ground. And as long as your gear is properly grounded (not always a safe assumption), that signal will just disappear. On the other end, a device expecting a balanced signal will take the flat signal of the ground, invert it (still a flat line), and sum it with the lead, which won’t change anything because a flat signal + anything equals that thing.
But it will be quieter than a true balanced signal because a true balanced signal will actually sum a pair of signal when canceling out noise. In practice this means a balanced signal is about 6db hotter than an unbalanced one.
But…
That should be the end of it, but there are actually two types of balanced connections. “Balanced” and “Differentially balanced”. When we talk about balanced connections, we usually mean “differentially balanced”. This is the kind where there’s an inverted copy of the signal on one of the lines. Regular plain “balanced” is the same except it doesn’t send an inverted signal. Instead it just “catches” noise to be inverted and sum’d out at the end. This type of balanced connection does not boost signal by 6dbs. So you really can’t rely on balanced connections being louder or unbalanced connections being softer. It all depends on the implementation.
Here’s the thing, though. TRS cables are also commonly used to carry stereo signal (tip = left, ring = right, sleeve = ground). If you take a cable carrying a stereo signal and plug it into a thing expecting a balanced signal, what happens? It will take the left and right signals, invert them, and sum them! This will usually cause the signals to cancel out and result in near silence or bad phasing.
(TRS is also commonly used for “insert” effects on mixers where the tip is the send and the ring is the return. Mixing this up with something expecting a stereo or balanced connection will obviously screw things up spectacularly!)
So there’s no simple rule for when to use TRS or TS because they’re all used for different things. In general, if you use TRS for all things expecting stereo (on both ends!) and TS for all things mono, you won’t get any benefit from any balanced connections your gear happens to support, but everything should behave as you expect. But ultimately, there’s no substitute for diving into manuals, laboriously drawing out what ports expect what, and making sure all your wires are providing that. 