Electronic – How more accurate in practice will a STM32 become if we use HSE

The timers divide the clock down by integer values. The long term average coming out of a timer will be the same as that of the oscillator.

It sounds like your timer period is off by 1. Read the datasheet carefully. You probably didn't put the right value into the period register. Usually these timers work such that the period is the period register value plus 1.

Another possible source of error is how the crystal is driven. The accuracy is specified for a particular load capacitance, which can often be a bit unpredictable since stray capacitance can be a significant part of the total. However, that would not explain why two timers running from the same clock don't agree on elapsed time. Only a firmware bug can cause that.

They are more accurate over a wider temperature range, they have far less jitter (cycle by cycle timing precision) and in general are more accurate to within very good tolerances of the required frequency. If you then are multiplying this reference by many times to get the final clock speed in your processor, any error or inaccuracy of the clock source would be multiplied too, possible giving huge offsets in real vs intended frequencies and larger than acceptable clock jitter during time sensitive things such as high speed synchronous communications or asychronous with timeouts.

External crystals (usually Quartz) or resonators (usually ceramic, have less awesome tolerances but are still better than internal RC oscillators) are quite cheap and can have very small footprints like the Murata Ceralock series ceramic resonators that I often use for my Atmel AVRs and even my Atmel Cortex M3 (to get 64 Mhz from 16 is annoying by the way, when you have to also get 48Mhz for USB!).

As a side note, I found that 12Mhz is a much nicer crystal/resonator frequency for USB (just x4) and for most clock speeds (like, 120Mhz is just x10), and I see it often on ARM based boards reference designs these days.