I've looked into both and the core difference I've found is that partnership = owning a portion of the aircraft (usually indirectly through equity in a holding entity, such as an LLC or a corporation), where flying club = joining a club that owns the aircraft, but not actually owning a portion of the aircraft.
The real world economics of what I found was that the buy-in for the partnership (in my most-recent case, a roughly a 1/5 interest in the plane since there were 4 other partners) was around $24k upfront (cash, no financing in my experience), then a set monthly maintenance fee to cover the expected maintenance and to build a reserve for some of the unexpected maintenance, and then a set (fairly low) hourly rate that more or less covered consumables and a portion of use-based maintenance (overhaul reserve, etc.). Partners are also subject to "capital calls" to fund expenses for which the partnership does not have enough money to pay. In a partnership, you'll want to have a lawyer review the partnership agreement (or whatever other governing legal documents exist between you and the other partners) in addition to taking whatever other steps you'd want to take when buying a piece of an airplane. If you want out, you've got to find someone to buy your piece of the partnership - whether that is your other partners or someone else (if someone else, your other partners will probably want to get to know them before jumping in bed with them in a business sense). So, your investment (if you want to call it that) isn't very liquid, but you share the plane with a small number of people and have some sense of pride of ownership.
Contrasted with the flying club I looked at most recently, there were maybe 12-15 members or so. The club owns the airplane, and not the members. You would pay an initiation fee to join the club (around $1,250 if memory serves, and they'd let you meet payments on that over a year), monthly membership dues (similar in purpose to the "maintenance fee" I described in the partnership above, but a lower amount since more members mean people were paying for the same "fixed" costs), and then a set hourly rate that was higher than the partnership hourly rate, but lower than the hourly rate of an aircraft off of the FBO line. Because you're "just" the member of a club and do not own a piece of the airplane, you leave a club just like you'd cancel a membership to the YMCA or Costco - tell the powers that be that you want out and you stop paying dues. Club members are subject to assessments for "excess" expenses that the club can't pay for (just like every year at any country club I've ever been a member of!). As noted above, it's not unusual for flying clubs to be not-for-profit entities, so there's some money saved on taxes on consumables and other expenses. But otherwise, it's not far removed from renting from the FBO in a big-picture sense - nicer airplane, fewer people in the "renting" pot, usually (not always) experienced pilots instead of students, and a slightly different economic relationship.
Others have probably had different experiences with both, but that was the result of my exploring both options here in Atlanta and in upstate NY pretty recently. So, hopefully this is at least somewhat helpful.
Edit: One more thing to mention - my flight school has a "flying club" in that you pay an extra $325 a year plus a $50 initiation fee, and then you get a discount on aircraft rental and CFI rates. Members are not subject to any other fees, assessments or expenses - just pay the annual fee, plus the hourly rental rate of the aircraft. It always felt less like a flying club and more like a business retention tool to me, and I think it can easily be distinguished from most other "traditional" flying clubs.