Both terms are something completely different and only intersect very little. To find the right term is actually very easy, just think about following:
A simulation does not do anything for real. You can study it, for example how computer work, but it usually has no outcome other than that. A plane crash in a Flight Simulator causes no real harm. A weather forecast simulation itself does not change the weather.
An emulation does something for real. You can work with an emulated computer like with a physical one and create documents with it. And a plane crash in a Flight Emulator would have an outcome, like people experiencing the real impact including possible physical harm.
Your confusion probably stems from the fact, that "studying the simulation" and "accessing the emulation" often is quite the same thing.
You are not alone with your confusion. The Film "Matrix" speaks of a simulation. However The Matrix is running an emulation, as it has real impact on all members of The Matrix. In contrast the training room has no real impact, so this is a simulation (of The Matrix).
Let's see some examples.
Simulated vs. Emulated Rain
Take a water hose in the garden and let it rain. What's the difference between simulation and emulation here?
When you are simulating rain, people still will blame you for getting wet. Your rain has some real impact on the world, but your simulation hasn't, as the simulation does not fool anybody in that it is real rain.
In contrast, when you are emulating rain, people would blame the weather. This is, your emulated rain really behaves like rain in reality.
This rain emulation hence distorts reality,
in making people belie in the wrong culprit.
It took me quite some time to understand that.
Hence it isn't easy nor obvious which explains all the confusion.
Keep in mind that a simulation can have sideeffects,
like the weather forecast is based on simulations,
which takes quite some computing power and thus electrical energy,
which has an environmental impact.
Hence in the example of "simulated rain", people getting wet just is a sideffect and not part of the simulation. Same is true if you simulate a rainbow with this simulated rain. While the property of "how rainbows work" is part of this simulation, the simulation itself is not providing the rainbow, this just happens due to refraction of the sun on the sideffect of the waterdrops.
Simulated vs. Emulated Computer
While you might think "a simulated computer can have an outcome" this is practically wrong reasoning. If you save files onto a simulated harddrive, these files cannot leave the simulated drive outside of the simulation. You can obtain the files by studying the simulated drive, but this is not part of the simulation itself.
In case the harddrive saves the data such, that the data is actually usable outside of the simulation, you have an emulated harddrive within the simulation to do so.
So an emulation can be part of a simulation and vice versa.
Simulated vs. Emulated Filesystem
If you simulate a filesystem, you probably, for practicability, will choose to save the files onto your real filesystem as-is (perhaps with some additional meta-information). In that case the simulation seems to create real "value" outside of the simulation: Usable files!
But this is just by coincidence, because your simulated filesystem actually emulates a filesystem as well. You actually emulated the outside filesystem inside your simulation!
Simulated vs. Emulated TPM or HSM
A good example of the difference is, when you think of security. A TPM is a specific device to keep it's own keys secure (source of identity) while an HSM is a general device to secure foreign keys (verify identity).
Fun Fact: My fingers constantly type TMP instead of TPM.
If you simulate a TPM this has a huge effect on security, because then you can observe the internal states of the TPM. Which renders all the security void. Even that such a simulation can give you valuable hints of improving the design of a TPM itself, you won't want to expose precious data to the simulated TPM for real.
However if you emulate a TPM you will try to hide these internal states to the outside as good as you can. Such an emulated TPM then can be possibly used to really secure something else better than without it.
With a real TPM you cannot emulate the properties of a real HSM. All you can archive is to simulate an HSM, but this will not have the security properties of a real HSM, so all data which is stored in this simulated HSM will not be protected (they will only be protected within the simulation itself).
In contrast, with a real HSM you can emulate a TPM with all properties of a real TPM. For this the HSM needs to be constructed such, that no information needs to leave the HSM which does not leave a TPM as well.
(Please note that I do not know anything about HSMs or TPMs in particular, so it might be that there are no HSMs out there which are able to provide emulated TPMs.)
Simulated vs. Emulated World
If our world is simulated, we are simulations, too. Hence some spectator (let's call her God) can look at us and change the simulation any time. Also we cannot find out if we are simulated or not. As I am pretty sure that I know that I am, I do not think I am simulated, because self-awareness looks like an effect with a real component to me, which contradicts simulation. This also means, our world cannot be a simulation, too, as a simulation can only affect me like the world does, if I am part of the simulation.
But our world still can be emulated (like in the Film "Matrix"), as all I have to "prove the world" is my state of mind and sensory input, which I cannot verify, as I cannot leave myself. If I am not part of the emulation, then there should be a chance to observe discontinuity (like in the film "Matrix"), in case the emulation does not work flawlessly.
This changes when I emulated, too, like running an OS in an emulator. Then I cannot observe such errors, as my state can be reset from within the emulation (call it: Sleep) without observable discontinuation.
However I rather think that the world is a holographic hallucination than something like an emulation. Because if it is emulated, then I am pwned by somebody (call him Rick) who is running the emulation for some purpose, while a hallucination is purely my own thing.
I stop here, because hallucinations lead us to something completely different.