Energy: The Bohr model and the Quantum Mechanical model of the atom both assign specific energies to an electron. In the Bohr model the energy of an electron is determined by the Rudberg equation and depends only on n, the principal quantum number.
The energy of an electron in the Bohr model corresponds to a specific and fixed distance from the nucleus. In contrast, the Quantum mechanical model treats electrons as waves mathematically. Schrodinger's wave equations are complex mathematical models that describe the energies of the electrons. As with the Bohr model the energies of electrons are quantized, or have only certain allowable values.
Schrodinger's equations are based on four quantum numbers. This changes the Bohr model to one that is three dimensional and has energy sub-levels (s,p,d,f) for each principal quantum number. The Bohr model treats the electrons with the same n value as degenerate, that is, having the same energy.
Positions Occupied by Electrons: The main similarity between the two models is that in both electrons are different distances from the nucleus, corresponding to different energies. The Bohr model can be visualized as similar to a solar system in which each electron is a specific distance from the nucleus with a known angular momentum related to speed. The allowable distances are stable orbits in which angular momentum doesn't deteriorate as classical mechanics predicts it should.
The Quantum Mechanical model doesn't specify exact positions of electrons. The solutions to the wave equations produce three dimensional areas of probability of finding an electron. In the Bohr model electrons with the same principal quantum number are the same distance from the nucleus, while the sub-levels of the QM model have different average distances from the nucleus. The Heisenberg Uncertainly Principle mathematically describes the uncertainty of the position of an electron in the QM model as being inversely related to the uncertainty of its momentum.
Shielding or Screening Effect on Outer Electrons: Bohr didn't incorporate the idea of shielding in his atomic model. He described only the hydrogen atom, to which the concept of shielding doesn't apply because it has only one electron and one energy level in the ground state.
In the QM model, the force of attraction to the nucleus by outer electrons is diminished or "shielded" by the repulsion of electrons closer to the nucleus. Each electron has a shielding factor related to the number of electrons between it and the nucleus. The shielding factor decreases the effective nuclear charge and is related to periodic trends such as atomic size and attraction for electrons.
The only similarity with respect to shielding is that if the Bohr Model is applied to multi-electron atoms it it will imply shielding since electrons repel each other.