Both are implementations of exactly the same concept, even up to the units of the tuning constant being seconds (mm/(mm/s)). The difference is in the implementation details, and particularly how they deal with a mathematically and physically nasty/demanding part of the concept.
In order to compensate for pressure making the filament/filament-path behave like a spring, LA/PA offset the E-axis position by a fraction (K) of the pre-LA/PA E-axis velocity. Mathematically, as an operator acting on the function that's the E-axis position at time t, the basic LA/PA transformation is:
I + K*D
where I is the identity, K is the spring constant, and D is the (time) derivative.
Anyone with a mathematical background in functional analysis or PDEs will recognize this as an unbounded linear operator (at least with respect to most norms, due to the derivative term). Casually speaking, it shifts bad behavior of a particular derivative "down one level". Instantaneous changes of acceleration (starting to accel/decel) become isntantaneous changes of velocity. Instantaneous changes of velocity ("jerk"/junctions) become instantaneous changes of position. (Insert horrified face here.) In short, the result is not physically realizable.
There are two ways to deal with this impossibility:
The Marlin way (Linear Advance): E-axis acceleration and jerk limits are applied in such a manner that the result of the LA transformation is still within those constraints. This makes printing a lot slower. Essentially there's no longer any E-jerk, and E-acceleration is limited by K time E-velocity.
The Klipper way (Pressure Advance): No changes are made whatsoever to the kinematic timing. Print speed is exactly what it would be without PA. Instead, a smoothing convolution with a triangle wave is applied on top of the result of the PA transformation (I + K*D) to make the E-axis position function back into something physically realizable. This introduces a new type of error into the extrusion, but if the time window of the smoothing can be kept narrow enough, the error is expected to be small enough not to matter. Unfortunately, the more extreme your print acceleration, the longer the window needed to keep the result of PA physically realizable, so there are still limits. If you don't tune them right, the extruder will just start skipping.
In my opinion, the Marlin behavior is easier to get started with, but hopelessly slow to actually use unless you start cranking up your acceleration and jerk limits. And this ends up being comparable to the manual tuning you have to do with the Klipper PA smooth time window. But in the end, Klipper's approach will always win on print performance, because the smoothing lets you cheat and do moves that wouldn't be possible with Marlin.
