Vbeon must be specified at some value of colector current & some temp, usually 25C. If the bjt is in its active region, i.e. Vce>0.7V or so, then Vbeon is the nominal forward voltage drop across the b-e junction.
To saturate the device, a base current greater than Ic/hFE is needed. Most general purpose bjt devices spec saturation at Ic/Ib equal to 10. In this case, the base is overdriven so that Vce drops to a low value, known as Vcesat. This is a condition where the device acts like a closed switch.
Because the base current is higher than its value when the device is in the active region, Vbe is higher as well, & this is Vbesat. Since Vbesat occurs when base current is increased, then Vbesat>Vbeon. But the voltage across a forward biased p-n junction increases only as the logarithm of the current, a large increase in Ib results in a relatively small increase in Vbe. Hence Vbesat is larger than Vbeon by a small margin.
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Claude