John and Jonathan do not share the same root name, and they are not basically the same name. They are not comparable to Sophie/Sophia, for example, which are clearly from the same roots.
John comes from Iohannes, the Latin form of Ioannes, which is the Grek form of the Hebrew name Yochanan. It means God is gracious.
Jonathan comes from Ionathan, which is a Latin and Greek form of the Hebrew name Yehonatan (or simplified as Yonatan). It means God has given.
Any source which says they share the same roots, are incorrect, but hey, I'm a Hebrew speaker, so what do I know?!
To give examples of names which do share the same Hebrew roots, James and Jacob do, as do Mary and Miriam.