I googled a bit...

Philips:
MV up to 125W incl. - outer bulb is made from the soft glass (the same with other manufacturers)
MV 250W and above - outer bulb is made from the hard glass (the same with other manufacturers)
ML (SBMV) 250W - both soft and hard glass (xxxSG/xxxHG versions)
Outer bulb temperature 350 Celsius is specified for all of them. This is logical as the bulb size grows simultaneously with the wattage.

Osram:
MV up to 125W incl. have their outer bulb temperature 310 Celsius.
MV 250W and above have their outer bulb temperature 350 to 380 Celsius.
SBMV up to 380 Celsius.

I wonder why it is so. If there are no relevant differences in temperatures, are there any construction reasons? Or better resistance against thermal shocks (burning lamp sprinkled by water etc.)? Better UV safety with high wattages...? Thanks.

Larger bulbs means larger bulb, so more stress with the same acceleration (vibration,...).
So with the same materials, a larger bulb would have to be made of a thicker glass. But a thicker glass has two consequences: The bulb gets heavier (so even more stresses) and it becomes more difficult to fabricate it without internal stresses weakening the structure.
So it becomes beneficial to use a stronger material. The higher strength allows to keep the glass wall thinner. And as the hard glass has lower coefficient of thermal expansion (well, that is the main reason, why it is stronger), even when thin, it is still rather strong.

Small bulbs does not make as much sense to fabricate from the hard glass (unless extra strength is required), as the glass technology does not allow to make the bulbs with thinner walls, so it would be more expensive, but it couldn't be made lighter.

When I opened a MV
 
I was surprised by a strength of the implosion. (I had had the outer bulb wrapped by a sellotape so the inner burner wasn't damaged.)

The vacuum (low pressure) inside the outer bulb (at least when it's cold) can also be the reason for making bigger ones from a hard glass to prevent them from "deflating".

There is just a partial vacuum in the outer, a carefully chosen pressure to provide just enough cooling required for the arctube to operate correctly. Well, with 50W it does mean really a vacuum, but with kW beasts nearly an atmospheric pressure. The fill gas is usually Nitrogen, as it is quite robust against flashover.