They are actually two different names. John is the Hebrew name Yohanan, or God is gracious, while Jonathan is Yonatan or Yehotanan, or God has given.
Spelling nit: Jonathan has been the standard spelling in the English language. I too have read what Gama_Rex avers, that they are independent names, but I don't have a source.
For example, from US Social Security Administration data at http://www.ssa.gov/OACT/babynames/#ht=2 , Jonathan has been in the top 100 in the US since 1900, in the top 100 since 1962, in the top 20 reliably until 1991. Johnathan with an H is unattested before 1952 and has never been in the top 100.
(If you want some medieval and Renaissance attestations, you can check articles under http://www.s-gabriel.org/names/ and http://heraldry.sca.org/names.html . In general, Old Testament names tended to be less popular in England before the break with Rome or so, and of course there are more records from later, so you might prefer to look at 1500s data.)
Supplementary question ... Is the contraction Jno. -- seen in many 19th century documents such as the census -- derived from Jonathan, John, or something else?