[Jon Clark]

I just read that for a H-atom, all orbitals in different subshells of an orbit are degenerate, and have same energy. Why does this happen in H-atom and not in other atoms?

[Corribus]

What do other atoms have that hydrogen does not?

HINT: Google "hydrogenic atom".

[Jon Clark]

A hydrogen atom has a single electron. So a 1s subshell is enough. Is this the reason that other orbitals are degenerate?

[Corribus]

The orbital (electron) energy is basically determined by the energy of attraction between an electron in the orbital and the nuclear core charge. In a multi-electronic atom, the effective core charge felt by an electron is influenced by other electrons in the system, particularly those that are closer to the nucleus. There is also an effect called "orbital penetration", which is more pronounced for some types of orbitals than others. For the same principle quantum number, n, S orbitals have more electron density close to the nucleus than P (or D or F) orbitals, an effect that arises from the quantum mechanical (wavefunction) treatment of electrons. Because S orbitals "penetrate" closer to the nucleus than P or D or F orbitals, they feel less shielding of the nuclear core charge by inner electrons, and thus have lower energy. This explains the "fill order" of orbitals in multielectronic systems. In a hydrogenic atom, there is only one electron, so the penetration effect doesn't matter.

https://www.youtube.com/watch?v=VWza-PlQTaU

[Jon Clark]

Would this happen in atoms like Helium also?

[Enthalpy]

Helium: if it's excited to 1s¹ 2s¹ versus 1s¹ 2p¹, the energy will differ, for the same reason as for Li 1s² 2s¹ versus 1s² 2p¹. A nice representation of orthohelium and parahelium there
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/helium.html
where the spin alignment of the electrons differs, and the energies of both diagrams have the same scale. It illustrates that the exclusion principle isn't the complete story.

Hydrogen 2s¹ and 2p¹ has the same energy for most people, but the tiny difference was observed experimentally to confirm the "Lamb shift" predicted by Lamb, which... err... well, check by yourself there
https://en.wikipedia.org/wiki/Lamb_shift