"Assyria" is the Latinization (via Greek) of the Akkadian/Aramaic Aššur, or Ashshur - the chief deity and divine monarch/figurehead of the Assyrian empires, as well as the name of the capital city of Assyria in most of its iterations. As I'm sure you know, Assyria was predominately focused around what is now northern Iraq, although it extended at times as far as the Mediterranean and the Persian plateau.

"Syria", referring to the geographical region in the north of the Levant, comes from the Greek Σύριοι/Syrioi, a shortened form of Ἀσσύριοι/Assyrioi ("people of Assyria"), and was used indiscriminately to refer to both Assyrians-proper in Mesopotamia and the inhabitants of the land known locally as Aram (at least until the Arab conquest in the 7th century CE).

The ambiguity exists as far back as Herodotus (7.63.1):

οὗτοι δὲ ὑπὸ μὲν Ἑλλήνων καλέονται Σύριοι, ὑπὸ δὲ τῶν βαρβάρων Ἀσσύριοι ἐκλήθησαν.

They are called by the Greeks Syrians, but the foreigners called them Assyrians.

Apparently the Syrioi abridgment in Greek comes from an earlier Luwian form, Surai.

So yeah, basically, they are related, as separately-evolved exonyms derived from the legacy of the mighty kingdom of Aššur. I wish I could include an example the Akkadian cuneiform here, but I can't figure out the unicode or find an image. Of course, it changed over time just as all terms do - it started out as AN.SHAR in Sumero-Akkadian I think, meaning "Whole Heaven", and was written in a few variations of that root over the centuries.