Physicist: Probably not.
Tiny things don’t act the way they should. You got super-positions, wave-like behavior, action at a distance, quantum tunneling, quantum teleportation, interaction-free detection, all kinds of things that should be impossible or paradoxical but aren’t. The laws of the very small (quantum mechanics) work well on their own, and the laws of the large (classical physics) work well on their own as well. The difficulties (and most of that long list of weird effects) come from the interplay between the big and small laws. For example, in quantum mechanics things can be in multiple locations, but in classical physics they clearly can’t. When something in multiple places (small rules) is detected by some kind of detector, it’s always found to be in one place (large rules). The Copenhagen and Many Worlds (MWI) interpretations are attempts to explain why there seem to be different, contradictory rules for large and small scales.
The Copenhagen interpretation basically says “large/complex things are always in one state, don’t obey quantum mechanical laws, and *something* happens when small quantum mechanical systems interact with larger systems that forces them to be in one state as well”. Copenhagen comes in a lot of flavors; consciousness-based (which can come with a side of Chopra or Secret), size-based, complexity-based, random collapse, and so on. The most common form of Copenhagen (in non-spiritual circles) is a size and/or complexity type: for some reason, when things become large enough, or interact with things that are large enough, they stop behaving quantum mechanically.
The advantage is “somehow” is a pretty safe thing to say, and (as long as you don’t look at the details) Copenhagen is also the simplest interpretation. However, there’s no end to the variety of contradictions and paradoxes it creates. If you’ve ever heard of a quantum mechanical paradox: that’s one.
The Many Worlds interpretation makes a different safe, but weird, statement, “quantum mechanical laws, all of them, work all the time at every level and the large-scale, classical world we see is actually a result of those laws”. MWI also comes in a few flavors, but the “keep using the same laws” version is the simplest to say, and most difficult to understand. The exact reasons behind why quantum mechanics leads to a classical world are subtle and math-heavy. In quantum mechanics not only can individual particles be in multiple states, but so can groups of particles, and even systems of arbitrarily great size and complexity.
The advantage is that all of those paradoxes that come from trying to reconcile two different kinds of physics (quantum vs. classical) disappear. On the other hand, when you try to describe some of the repercussions you’ll sound balls-out crazy. The same “many-stateness” that shows up for individual particles and tiny systems shows up for everything including people, puppies, countries, solar systems, freaking everything. In exactly the same way that a single photon can pass through two slits, or otherwise be in a tremendous number of states, a person can live an arbitrarily large number of different lives all at once (the different versions are “unaware” of each other in both cases). Each of those different versions experiences a different classical (seemingly non-quantumy) world.
Unfortunately, both MWI and Copenhagen predict that the world of the small and the world of the large will behave differently (or at least will appear to behave differently). So (to finally answer the original question) the two interpretations differ in their experimental predictions only for larger systems. While MWI says that things will continue to obey quantum mechanical laws forever, Copenhagen says that at some point between the atomic scale and the every-day scale a system will have to obey only classical laws.
Back in the day, when these two interpretations were first conceived, the only things that had demonstrated super-position were individual particles. However, since then intrepid, handsome/beautiful physicists have managed to demonstrate the same quantumy properties in progressively larger things, including buckyballs and even a metal, 30 μm long needle large enough to be seen with the naked (squinting) eye.
Every experiment that can be done, so far, has returned a positive result for the quantum nature of things, and pushed back the “Copenhagen scale” at which classical physics must take over. Considering that the scale at which Copenhagen would have to kick in keeps getting larger and larger, it seems like MWI is the better choice. The goal posts to disprove Copenhagen get moved farther and farther back by fancier and better controlled experiments. Copenhagen is fast becoming a “theory of the gaps“.
A really, completely, absolutely definitive experiment would need to demonstrate some form of super-position with an object as big as (or involving) a person. However, the larger something is, the more difficult it is to detect its quantum nature, and the more carefully controlled the experiment must be. Ideally, putting someone in a box, demonstrating that that person can behave in a “many-state-kind-of-way”, and then bringing them out would put the debate to rest. Unfortunately, for a number of reasons, this experiment is very unlikely to ever be done.