Warm up is the main reason. For the comparison with cars, the only thing
cars and buses have in common is that the engine works the same way yet
that's about it. AFAIK the biggest diesel in cars was the Mercedes 300D (6
cylinder 3.0 litres) the most common are about 2 litres now. The most
common bus engine will be around 10 to 12 litres (the smallest about 7
litres going up to 14 in coaches). These things do behave a little
different, especially when cold.

Yet there could be another reason: additional heating system (like Webasto
or Eberspächer). These tend to smoke quite a bit too.
As for that I've recently seen a car smoking like hell from the engine
compartment (wheel housing to be precise) but it was certainly not from
the engine itself. From what I know it looked more like heating.

The engines in modern buses now such as the Volvo B7TL or Dennis Trident or
even the Mercedes Citaro bendibuses tend to be quite small at 7 litres, a
move away from larger engines. The mini buses such as Darts have even
smaller engines such as 5.9 litre for the older ones, to 3.9 litre for the
newest Dart SLF. But yes they can be a bit smoke on start up. Once running
however I dare any car driver to floor his (diesel) car against its
rev-limiter and not be scared of either the engine falling to bits or large
amounts of smoke coming out. Mind you my coach won't rev fully in neutral
against the rev-limiter. Are cars protected in the same way now?