The cylinder (usually) has an electric immersion heater inside it. This screws in. It can be removed and replaced with a new one if it goes wrong. I've seen them still working after 50 years. The thermostat that fits inside it is also replaceable (without needing to drain any water). They last less long.
The cylinder (usually) has a copper tube in it, coiled. The boiler heats the inside of this coil, and the heat passes to the tapwater in the cylinder. The coil (almost) never goes wrong. If it does, you buy a new cylinder.
Modern cylinders come pre-insulated, and have a bigger coil than old ones so they heat up faster (modern boilers are usually much more powerful than 50-year-old ones so can supply more heat). They are sized so that a powerful boiler can notionally take the cylinder from cold to hot in 20 minutes (but in practice they usually take longer).
The modern unvented (high-pressure) cylinder is usually chosen to be larger than the old ones, so it can run at least a bath and a couple of showers, and a day's washing up, without running cold. But as they heat up quickly, by the time you've run your bath, wallowed in it, got out, towelled dry, and brushed your teeth, it will be replenished.
It is a bit more economical on gas to heat it once or twice a day rather than continually, because the boiler runs less often. In winter it doesn't matter, if the boiler is running anyway to do the radiators.