I may be predicting more than telling, as I haven't gotten anywhere near residency and am simply compounding the stories of other residents into an analysis; so take this as perhaps not universally true:
Being 'in' clubs does nothing for you, though it makes you feel good to do. 'leading' clubs does nothing for you, except show interest in a field, if you only do things within your school and its community. Being in a club and attending meetings lets you shake some hands and while it still does *nothing* for your CV, you may know some PDs or physicians ahead of time and those specific people will give you better LoRs.
Now if you are on a club leadership, go to national meetings, and you hold a national position (or if your club is REALLY prominent, a state position) you can really make an impact on your CV. I want to differentiate on this one a bit to make sure I'm clear. If you have a national position in any club you can really make huge inroads in that field. That's clear. Now as to what groups I think a state position is enough for? Pretty much only state medical societies (AMA sub-branches) and State Osteopathic Societies (not neccessarily SOMA, I mean AOA state branches). Just wanted to clarify that even if you're at the state (or national) level, being stuck in the student section 100% of the time doesn't make an impact.
I'm personally greatly involved in the state and national level of the AMA, holding state positions. It lets me talk to tons of students and representatives from all different groups (since every group known to man and not included in the AOA has representation at the AMA events) and it seems that those who make their impact in specialty societies seem to feel that reaching the level they did flipped a binary switch between stuff they did just for fun, and stuff finally paying dividends within their field. AMA/AOA national involvement does the same thing, but I am getting to know PDs from every field really well and keeping in contact with them after the meetings. I think you really need to rise to such a high level to make a difference, anything less than that is just "oh he is interested" but is rather insignificant.
TL;DR version: Clubs in medical school are basically self-stimulation. Feels good to do, but doesn't accomplish the actual goal it was made to get you (a leg up on residency). The exception here is holding national positions (attending isnt enough), and while both are effective; AMA/AOA > Specialty Society.