What's the difference between 50Ω and 75Ω connectors?
The impedance (the electrical characteristics), and that's created by the dimensions of the parts (sizes and spacing between them), and the materials they're made out of (including the insulators).
The two are supposed to be mechanically interchangeable. i.e. A 50Ω plug is supposed to fit into a 75Ω socket, and vice versa. But that doesn't always work, though that would be a manufacturing defect (some bad manufacturer not building the parts to proper specification, simply building their own plugs and sockets to fit each other). Leaving you with two choices; buy good quality parts, or buy your plugs and sockets from the same manufacturer. In fact, sometimes the only way to get plugs that fit the cable is to order them from the same supplier and specify that they must fit each other. Not only to deal with manufacturing tolerances, but also that solid core and stranded core cabling requires different pins inside the BNC plug (if they're crimp-on pins), and you may want to choose crimp-on or solder-on pins. You may even have to order a specific crimping tool to get a good fit.
But the plugs and sockets for the two impedances are not the same on the wiring sides. 50Ω and 75Ω cables have different dimensions (both the inner core, and the outer sheath), and they each have different guages of cable, as well. So the connectors are made to suit, and the wrong one will not fit right, if at all. You will need to buy the right connector for the cable it's going to be fitted to. And vice versa (e.g. screw-on, and most crimp-on, connectors need cabling with a solid inner core).
Well made crimped cables can easily last a couple of decades of use (being plugged and unplugged, being flexed and dragged about). They're only going to fail if abused, poor quality cables or connectors are used, badly fitted, or incompatible parts are used (mismatching cable and connectors).
The adjacent photo shows a couple of well made crimped cables that are around twenty years old (disconnected) and a couple of badly made ones (connected). The furthest to the back, and plugged in, connector has stray wiring protuding between the crimped collar and the plug. That's not really much of an electrical problem, more of an issue with getting scratched and cut fingers. The nearest to the front, and plugged in, connector either hasn't been crimped over the top of the insulation, or the insulation has slipped out of place. This may become unreliable, will become unreliable if it has no grip, and is more prone to having the connector ripped off if the cable is dragged across the floor, and it snags on something.
Badly manufactured parts may: Have pins that are too big or small for the mating socket, causing intermittent connection or damage to the socket. Wrong sized dialectric (the, typically, white plastic in the middle), leading to sloppy fits or connectors that won't fit together. Pins that don't properly fit the wiring inside the connector, that come loose. Wrongly sized outer crimp rings that don't fit over the coax, or are so sloppy that they don't grip it.
And you need good quality coax to be able to reliably crimp a connector on. All parts of the cable has to have the right dimensions, and a fairly tough braided shield is required. Foil shielding is going to rip off, likewise with thinly braided shielding with only a few sparse wires.
75Ω cable is traditionally used with television video and RF leads. The usual thinner cable being RG59 (that's a military cable specification, not a particular product code; theoretically it means that any cable identified as RG59 should have the same characteristics as any other RG59 cable, though some cables may actually be better, and some may not actually match the specifications properly), commonly used for patch leads and other flexible leads on mobile equipment. And the usual thicker cable being RG6, used for higher quality, low loss, less flexible cabling, such as rack mounted equipment that's fixed into position, and used with good antenna wiring.
50Ω cable is traditionally used with communication equipment (radio antennas and computer networking). The usual thinner cable being RG58 (it's smaller than RG59), and the usual thicker cable being RG8 (bigger than RG6).
I say “usual” above, as there are many different guages of cable, but those are the ones most commonly encountered in domestic and some industry circles. Much bigger and better cables will, usually, be used with professional transmitter wiring.
For short runs, and non-transmitter wiring, you won't really notice any difference between the two. Both will work fine for receiving antenna wiring, and video cabling. You might find that some of the cheap 50Ω coax is a better build quality than some of the common, cheap and nasty, 75Ω coax. But for longer runs, where cable losses become quite significant, and impedance matching becomes more critical, there is a difference.