What is the essence of the Hanbury Brown and Twiss effect?

Wikipedia is talking about two ways of interpreting this effect.

The classical way is simply calculating the intensity-fluctuation-correlation. But what is here the difference between a laser and a thermal source?

For the QM interpretation they refer to the two-photon interference. What is this?

As I understand:

Thinking of a thermal light source which we filtered (frequency and spatial wise) to get almost coherent light – there is still a big difference to a laser. The statistic! While the number of equally emitted photons from at laser is poisson distributed – it is Bose-Einstein distributed in the thermal case.

But how is this connected to what wikipedia is telling us?