Use the same booster and the sound changes completely with where you place it.
Before the amp, or after the distortion — in the amp's loop. In circuit terms it's just a difference of position, but the result is an entirely different thing. Why does position alone change so much? Sort this out and your sound-making gets clearer.
Placed before, it changes how the distortion comes in
What happens placing a booster before the amp — that is, before the distortion?
The most famous example is inserting a Tube Screamer (Ibanez TS808/TS9) in front of a high-gain amp. Placed ahead of a distorting amp like a Mesa/Boogie, Peavey 5150, or Marshall, it adds little gain but trims the lows and lifts the mids before sending it in. Then the amp receives a tighter signal and makes a tight, pushy distortion. The staple metal and hard-rock sound is often made by this combination of "prep with the TS, then distort at the amp."
So a booster placed in front works not as a volume-raising device but as a "device that changes the material of the distortion." The quality of the distortion itself moves. This effect is what many players expect from a booster.
Placed after, it pushes out volume and range
What happens placing the same booster — or a clean booster (a low-color one like an MXR Micro Amp or Xotic EP Booster) — after the distortion, in the amp's effects loop?
The distortion is already made. Boost there and the quality of the distortion doesn't change much. Instead, the whole finished sound is lifted. Volume rises, it comes forward, it cuts in a solo. Stepping on a post-loop booster at the head of a solo to push one step forward is a typical use.
A booster placed after becomes a device that "pushes out the distortion you made," not "makes the distortion." Even the same pedal, its role swaps with where it sits. At the front, material; at the back, finish.
Position is the design of signal order
What comes into view here is that sound-making is decided not only by "what you use" but by "in what order you arrange it."
A signal is processed in the order it passes. A sound distorted after being pushed at the front, and a sound pushed after being distorted, have different histories. Even the same parts, a different order of passage gives a different sound. Position is the act of designing the signal's history.
This is an extension of the same story as modern high gain crafting the input stage finely. What, in what order, in what state you send into the amp. The design of position is the design of the input signal itself.
So position and all, it's one sound
Bring this viewpoint into recording and one conclusion falls out.
To keep a sound that used a booster, "used a booster" isn't enough. Before or after, at what setting, in what order it passed — including that, it finally becomes that sound. Drop the position and you can't reach the same sound.
Even the same parts, a different order is a different thing. If so, order is part of the sound.
In short
Even the same booster, placed before it changes the material of the distortion; placed after it pushes out volume and range. A difference of position alone gives an entirely different sound.
It's the design of the order the signal passes, and part of the design of the input signal. When you add one pedal, whether to place it before or after isn't a trivial matter of wiring. It's a design decision itself — what sound you want to make.