The problem is you are changing seetings on your editor itself rather than the object you want to edit. You need to move all the properties over to Weapon and make them public. Then use the target property of the editor to get at the instance of the Weapon you are editing.

public class Weapon : MonoBehaviour {
    public string myString = "Hello World";
    public bool groupEnabled;
    public bool myBool = true;
    public float myFloat = 1.23f;

    void Start()
    {
        Debug.Log(myString );
    }
 }

[CustomEditor(typeof(Weapon))]
public class MyWindow : Editor {
    public override void OnInspectorGUI()
    {
        Weapon weapon = (Weapon)target;

        GUILayout.Label ("Base Settings", EditorStyles.boldLabel);
        weapon.myString = EditorGUILayout.TextField ("Text Field", weapon.myString);

        weapon.groupEnabled = EditorGUILayout.BeginToggleGroup ("Optional Settings", weapon.groupEnabled);
        weapon.myBool = EditorGUILayout.Toggle ("Toggle", weapon.myBool);
        weapon.myFloat = EditorGUILayout.Slider ("Slider", weapon.myFloat, -3, 3);
        EditorGUILayout.EndToggleGroup ();

    }
}