To be honest, I'm now a little stunned myself at how complicated the answer seemed at first, and how simple the solution is in the end.

[![How could I get the distance of a point to its nearest point - Animation 2][9]][9]

**Here we go:**

1) First create your points with `Distribute Points on Faces`.

    [![How could I get the distance of a point to its nearest point - Step 1][1]][1]

2) Then use the node `Convex Hull` to connect the points with a hull. This step is helpful because it gives us a triangulated mesh to work with further. And since we're dealing with a sphere here, we won't lose a point with this technique

    [![How could I get the distance of a point to its nearest point - Step 2][2]][2]

3) Using the nodes `Extrude Mesh`, `Split Edges` and `Separate Geometry` you get the isolated edges of this mesh.

    [![How could I get the distance of a point to its nearest point - Step 3][3]][3]

4) Then reduce the scale of each edge by half.

    [![How could I get the distance of a point to its nearest point - Step 4][4]][4]

5) Now that the edges are reduced in proportion to their length, you can reliably find the nearest point with the node `Geometry Proximity`. If you then calculate the direction vector between your originally created points and the position results of `Geometry Proximity`, you will know in which direction the shortest vector points.

    [![How could I get the distance of a point to its nearest point - Step 5][5]][5]

6) In the last step you only have to correct the length. Since you have shortened the edges by *50%* before, you simply scale the direction vector by $4$, which is exactly the point you were looking for (Apart from a few minor rounding errors).

    [![How could I get the distance of a point to its nearest point - Step ][6]][6]

The final result is this (Each previously created point is here connected to the nearest point):

[![How could I get the distance of a point to its nearest point - Result][7]][7]

...and with animated *Seed/Density* it looks like this:

[![How could I get the distance of a point to its nearest point - Animation 1][8]][8]

Here is an overview of the node group:

[![How could I get the distance of a point to its nearest point - Node Group][10]][10]

Here is the blend file *(I added an additional view for debugging)*:

[<img src="https://blend-exchange.com/embedImage.png?bid=0XVj1Vav" />](https://blend-exchange.com/b/0XVj1Vav/)

...and as a bonus I added the animation to the blend file too, because it's so nice to see the thing in motion (even though I won't win any beauty contests with the node tree, but it's meant as a little animation example).

  [1]: https://i.sstatic.net/d0dul.jpg
  [2]: https://i.sstatic.net/vDkZT.jpg
  [3]: https://i.sstatic.net/XJyu9.jpg
  [4]: https://i.sstatic.net/GgDPV.jpg
  [5]: https://i.sstatic.net/7HEq0.jpg
  [6]: https://i.sstatic.net/zOBNa.jpg
  [7]: https://i.sstatic.net/yuvkz.jpg
  [8]: https://i.sstatic.net/1HW40.gif
  [9]: https://i.sstatic.net/kGjTd.gif
  [10]: https://i.sstatic.net/u3eD2.png