The "Run" mode is the typical running mode which allows for the highest clocks. It may have various core voltage levels applied to the core which can save a bit of power or achieve the highest clock speed.
If you don't need a high clock speed you can use low-power run. On most parts where I see this featured, the limit is usually around 2-4MHz or so. On STM32L4 and other parts, it switches to a lower power regulator. Similarily, this lower power regulator is also used for sleep durations. This mode is useful if you need the MCU to remain active for long periods of time without fast responses. The low power STM32s also feature a MSI oscillator which you can scale down to tens or hundreds of kHz without paying a large biasing current of a high-speed crystal/oscillator. I used this mode on STM32L4 to run the part at 100kHz for long durations.
In STM32 world, the term "sleep" is a bit confusing though, because it only refers to the CPU not executing instructions. The rest of the peripheral world keeps running as-is. Personally I would refer to this as an idle state, but whatever.
The various stop modes scales down the peripheral domain. FLASH is often disabled as it is not expected the CPU needs to quickly resume running code. High-power peripherals (like USB or ethernet) may be disabled as well, then high freq clocks and eventually only a limited subset of low frequency peripherals (e.g. a LPTIM) can remain active.
Then at the low end of operational modes is standby, which is probably the more classic meaning of 'sleep'. This mode typically only has brownout and RTC running, that's it.
The STM32U5 chip is a bit unique, as it features a Smart Run Domain in which it can request a clock on-demand for high-frequency peripherals, even in STOP0-2 (iirc it even features STOP3). This is a feature that was long overdue as MSP430 and SiLabs chips had this for quite some time.
By having clocks on-demand, there is less need for a LPRun/LPSleep mode that keeps high-frequency oscillators active. The code can 'race to sleep' again, which is typically a good strategy to save energy.
However if you're concerned with battery life, total energy draw (as opposed to power) are things to consider. High current spikes don't always help, so that could compromise the chosen clockspeed.