First, I wouldn't consider PBR to be separate from forward or deferred rendering. You can use PBR in either forward or deferred rendering, as I understand it. Physically-based rendering just means trying to simulate the actual physics of light. This gives more realistic results than other techniques, but like all techniques, it involves trade-offs between speed and accuracy.
I think your understanding of forward deferred rendering is correct. One issue you can run into with forward rendering is that there is a limit to the amount of data you can send to a single shader. If you have 100s of lights, you need to send the position, direction, type, color, etc. for each one to the shader. You can hit the limits of number of allowed uniforms in scenes with lots of lights. It's also possible to hit the limit of how much work can be done in a given draw call. I've had the OS kill drawing when a single shader took too long. (This was not with a game, though.) With deferred rendering, you simply make multiple passes sending a subset of the number of lights to each pass. Note, though, that every pass you add adds a lot of overhead to the rendering, making it slower.
I don't see how Forward and Deferred render are going to give much difference to the scene.
The results of forward and deferred rendering may in fact be the same if you're sending the same numbers of lights and rendering them the same.
Why do we need to store the geometrical value in Deferred Render before applying lighting? Why don't we just compute lighting right away like Forward Render?
If you computed it right away, it wouldn't be deferred anymore. That's the difference between the 2. Deferred rendering renders the geometry without taking into account lighting, then in subsequent passes calculates the lighting. Forward rendering does it all at once. (There may still be multiple passes for other reasons, like depth-peeling, etc.)
Also for Physically Based Rendering, computing each light takes a lot of time, and almost impossible to be done in real time. But why this technique is always considered the best?
Computing each light takes more time than for something simple like Phong, but I wouldn't say it's "almost impossible to be done in real time," as there are plenty of games out there using it now. "Best" depends on your goals. If your goal is to run on low-powered hardware, PBR might not be best. Currently it gives realistic results, so it's easy for artists to understand (in that it matches the world they live in) and looks "right" to players. But it has limits, like any technique. It's been popular recently because it's been new and effective, when used appropriately.