What's in their lane
Code Compliance (sometimes called Code Enforcement) handles violations of a city or county's property maintenance code: overgrown or unmaintained lots, illegally posted signs, unpermitted construction, and junk or debris left on private property.
The dividing line is almost always public vs. private property. Public Works and Solid Waste only have authority over streets, sidewalks, and public rights-of-way — once an issue is sitting on someone's lot instead of the street in front of it, it's Code Compliance's problem, not theirs.
Where this overlaps with other departments
Illegal dumping on a public street routes to Solid Waste; the same pile of debris on private land routes to Code Compliance instead. Signs posted illegally in the public right-of-way are more of a Public Works or right-of-way issue; signs illegally posted on private property are Code Compliance's.
Graffiti works the same way — it's a code violation wherever it shows up, but who actually sends someone out can depend on whether it's on public or private property.
Reporting it
City of Miami's Code Compliance Inspector line is (305) 416-2087. Have the address ready — code violations are tracked and enforced property by property.