When you search in the internet, you hear a lot about failure of this this this classical mechanics laws, started quantum mechanics. But the real question is what actually defines which is classical and which is quantum mechanics.
At first glance, if you get any "n" in the energy equation, then that is quantum mechanics. Otherwise, classical mechanics. To understand this, first, you need to understand what quantum means. The word quantum, actually means any discrete numbers, like 1, 2, 3, etc. These should be positive, in this case. In simple terms, positive integer. So, you just need to look for (n=1, 2, 3..) to make sure that the energy equation you have is part of classical mechanics or quantum mechanics.
When you look at Wein's and Rayleigh Jean's law you will see something with frequency, wavelength, etc. But if you look at Planck's law, you will see the quantum number as n. This concludes the thinking that Wein's law and Rayleigh Jean's law are part of classical mechanics, but Planck's law is quantum mechanics.
Let's dive in a little bit. By looking at the graph of energy vs wavelength found from the Wein's law and Rayleigh Jean's law, you will see that the graphs are exponentially increasing or decreasing for changing the wavelength. Which is contradictory to the experimental findings.
(Related but not important) Graph of Rayleigh Jean's law of Energy wavelength indicates that if wavelength tends to zero then energy tends to infinity. This phenomenon is known as ultraviolet catastrophe.
So, Planck proposed some hypothesis with a law to fit with the experimental findings. In the law, he suggested a universal constant(h) named after him. This energy law consists of a quantum number, Planck's constant, and frequency.
Thus, started the world of quantum mechanics.