When do you use Jr. after a name as opposed to the II or III?
The designation of Sr. or Jr. to distinguish between father and son with all the exact same names (first, middle, & last), can be replaced by the Roman numerals, I and II, respectively, when the grandson has the exact same names. The grandson will then have a III after his name. The grandfather and father can continue to use Sr. and Jr., respectively, or the numerals.
However, using the II (not Jr.) often means a man does not have his father's name, but another relative such as his grandfather or uncle.
Royalty or other ceremonial title would always have the I or II, rather than Sr. or Jr. to designate the line.
When addressing the envelope with Sr., Jr., or Roman numeral, use first and last name and no comma. The salutation would not include the suffix.
Mr. John Smith II
Mr. John Smith Jr.
Dear Mr. Smith,
Please note that some may prefer to use a comma before the Jr. or Sr.
"Traditionally, it would be John Smith, Jr., and John Smith III. But beginning with the fourteenth edition of The Chicago Manual of Style (1993), the recommendation is to use no commas in either case (see paragraph 6.49 of the fifteenth edition" (from Englishrules.com, 2005).