That's probably because "CurrentSpawn" references a different object which is destroyed on load. When the new re-set version of the level comes around, that same spawnpoint gameobject is probably considered - by Unity - to be an entirely different new object.

Perhaps instead of storing a reference to that object, you should just store the position of the spawnpoint as a Vector3 (and also the rotation as a Quaternion, if that is required). Because Vector3's and Quaternions are structs, your script will retain its own copy of the values, rather than referencing some other object which gets flushed away when the Load occurs.

If you want to keep CurrentSpawn alive as well, you need to tell it not to destroy as well, by the way, to differentiate between vars and functions and objects its practice to start a variable with lowercase:

var currentSpawn : GameObject;

function Start() {
    currentSpawn.SendMessage("SpawnCharacter");
}

function Awake() {
    DontDestroyOnLoad(this);
    DontDestroyOnLoad(currentSpawn);
}

function Update() {
    print(currentSpawn.transform.name);
}

If an example would help, I posted an Answer in another Question:
How to Keep Information when Reloading a Level.
Which basically points to a demo package I wrote, with code, at:
Simple Demo of a Level Manager