They are heterographic as you say, but not synonymous. Since you already have the meanings pretty much figured out, there is no need to go into detail. To sum, 暑い describes weather/environment/temperatures, while 熱い describes objects.
When you hear or see 「あつい」, context often determines which word is used.
The Venn diagram below, copped from Wikimedia, demonstrates several related concepts. To be considered synonyms, the words have to have the same meaning but different pronunciations and spellings. Strictly speaking "synonyms" exclude homophones and homographs.
「暑い」and「熱い」fall square in the purple area.