Since you mentioned the Materials Project, I assume you are talking about computational results.
Formation energy is the energy with respect to some reference states. This is used for example with DFT results, since the absolute energy output by the program is not very useful without a reference. Energy above the hull is referring to the distance above the convex hull, which is formed by the lowest calculated states. In DFT, anything on the convex hull is considered a ground state, i.e. it is a stable state at 0K. For a simple binary system, you can calculate the formation energy by $E_f = E - (1-x)E_1 - xE_2$ where $E$ is the energy of the material and $E_1$ is the energy of the reference at $x=0$ composition and $E_2$ is the energy of the reference at $x=1$ composition. To get the energy above the hull, you have to know what the nearby stable phases are.
Here is an example. If you go to the Materials Project page for NaCl or open it in their phase diagram app, you will see the binary convex hull. The reference states are pure Na and pure Cl. Since they are the stable reference states, they will have a formation energy of 0 and also a hull distance of 0. NaCl also has a hull distance of 0 because it is a ground state, but its formation energy is nonzero. If you hover over the points, you will see the formation energy (usually negative) and hull distance (always positive) of each one. Note that formation energies can be positive, like on the Na side of the diagram, but usually people are looking for negative formation energies.
If you are comparing data from different databases, I think that the formation energy is probably safer because it only depends on the reference states and not on the ground states that that person found.
