If the application of the Bose-Einstein distribution is in blackbody radiation, then what is Planck's distribution? Are they same? How did Planck know that he should use a Bose-Einstein distribution to model blackbody radiation? Thanks
Planck didn't know Bose-Einstein statistics at the time around 1900. With the existence of minimal unit, or quantization $E=hf$, in mind, he derived the Planck's law which describe the black body radiation. Two decades late, after the establishment of the Bose-Einstein statistics, then it is known that Plank's law is a special case of Bose-Einstein distribution by simple using $E=hf$.
Plank's distribution (law) is a specific application of the Bose-Einstein distribution.
For example, there is no chemical potential, $\mu$, for photons, so it is missing from Planck's law, although it's in the Bose-Einstein distribution. (The chemical potential only comes into play when you have a fixed number of particles; there is no such restriction for photons.)
The Bose-Einstein distribution gives you a number -- the number of particles in a given state. Planck's law gives you a spectral radiance, which includes things like the number of states with a given energy (the "density of states"), and the energy of photons in those states. Those things are multiplied together with the Bose-Einstein distribution to give Planck's law.
In short, Planck's law contains the Bose-Einstein distribution, but it also includes other things.
These responses do not really answer the question. Planck's distribution was explicitly provided by Planck to represent the distribution of energy for what are now referred to as a boson. The proposition of Bose, 24 years later, was a quite terse reiteration of the same geometric expansion, and to recommend it as a more general principle for thermal distributions in other quantum systems of integer spin.
Saying that Planck distribution is "an application" of BE distrbution is intellectually dishonest. Planck developed the precise distribution factor of B-E for the precise purpose of modeling these types of quantum systems. He wasn't "applying" anything; he originated the science!
Why the BE factor is not actually called a "planck distribution" is a completely valid question, and completely up for debate.