It depends a bit how you define things. The Copenhagen interpretation has a collapsing wave function doing so in a way where we can't observe it collapse and the pilot wave theory has pilot waves all over the place that have no effect other than guiding the particle in question then effectively disappearing when not needed. If either the collapses or pilot waves were real things I would kind of expect them to be observable in some way, but in both interpretations they are not observable, which leads me to suspect they are not real things and will go the way of the "luminiferous aether".
Funny that you say that since the particles are pretty much all we do observe. You never "see" the wave function, but you do see where stuff is.
To be charitable to Copenhagen, we would say that we do see the results of collapse. The main problem with the collapse is that is simply not well-specified by the theory.
In pilot wave theory, one can deduce the standard quantum formalism. Operators, collapse, and all the rest just pop right out.
So no, there is no experiment that can tell the difference. It is possible that pilot wave theory has predictions that the standard formalism is silent on. However, something that is inspired by pilot wave theory can be easily co-opted by standard approaches, e.g., Bell's theorem.
As an example, it is very easy to place pilot wave theory on manifolds as these are just differential equations. For the standard theory, it is not at all clear what the main operators ought to be such as momentum. Position, yes, but not other things.
Of course, once one has pilot wave theory on it, then whatever needs to be deduced can be and then given to the standard theory.