AppleWorks started on the Apple II in 1984. Microsoft Works apparently was originally written by the same engineer after he left Apple (!!) but then licensed it to Microsoft, it came out in 1987.
Ah I found the wedge, still an apple office, and I had biked quite close to it at least once. On Patrick Henry Dr.
Clarisworks was a successor to AppleWorks for the Apple 2/2gs computers. Don't recall when MS Works launched but they didn't have an exclusive on the name. IIRC Appleworks was launched alongside ProDOS.
Also, I see a lot of places describing the Claris offices as 6 miles from Apple HQ, probably all from this source. Where was it exactly? I'd imagine in that commercial area in Sunnyvale and north Santa Clara, but I've biked around that area many times and never noticed a "wedge."