GameObject.FindObjectOfType<>() vs GetComponent<>()

There are two differences:

1.) GetComponent<T> finds a component only if it's attached to the same GameObject. GameObject.FindObjectOfType<T> on the other hand searches whole hierarchy and returns the first object that matches!

2.) GetComponent<T> returns only an object that inherits from Component, while GameObject.FindObjectOfType<T> doesn't really care.

You don't want to use

void Start () {
    levelManager = GameObject.FindObjectOfType<LevelManager>();
}

that often. Particularly on start

To answer your question though, these two functions are actually not very similar. One is a exterior call, the other an interior.
So what's the difference?

1. The GameObject.FindObjectOfType is more of a scene wide search and isn't the optimal way of getting an answer. Actually, Unity publicly said its super slow Unity3D API Reference - FindObjectOfType

2. The GetComponent<LevelManager>(); is a local call. Meaning whatever file is making this call will only search the GameObject that it is attached to. So in the inspector, the file will only search other things in the same inspector window. Such as Mesh Renderer, Mesh Filter, Etc. Or that objects children. I believe there is a separate call for this, though.
   Also, you can use this to access other GameObject's components if you reference them first (show below).

Resolution:

I would recommend doing a tag search in the awake function.

private LevelManager levelManager;

void Awake () {
    levelManager = GameObject.FindGameObjectWithTag ("manager").GetComponent<LevelManager>();
}

Don't forget to tag the GameObject with the script LevelManager on it by adding a tag. (Click the GameObject, look at the top of the inspector, and click Tag->Add Tag

You can do that, or do

public LevelManager levelManager;

And drag the GameObject into the box in the inspector.

Either option is significantly better than doing a GameObject.FindObjectOfType.

Hope this helps