It turns out that Mike Attisano, whose new travel guide, Simplifying Las Vegas, is available on Kindle and as a pdf at simplifyinglasvegas.com, did some research on the sign and graciously provides this answer.
The famous "Free Aspirin & Tender Sympathy" was manufactured by YESCO (Young Electric Sign Company), which has been lighting up Las Vegas since the 1920s. YESCO built Vegas Vic (the subject of many QoDs), and icons for the Stardust, Circus Circus, the Silver Slipper, and many more casinos. According to Ed Tomchin’s article "Honorable Discharge," YESCO’s Dick Porter designed the sign, which was erected in the early 1950s in front of a Union 76 gas station on the west side of the Las Vegas Strip just north of Tropicana Avenue.
On one side (facing south), the sign offered the famous phrase in large letters, which lit up at night. On the other side, facing north, the "free" information was smaller, and in larger letters a minimart advertised cold drinks, snacks, and film (in part for taking pictures of the sign).
In 1996, the Monte Carlo opened near that location on the Strip. Not long after that, the Union 76 was torn down, along with the sign, and a CVS took its place. However, before the sign was dismantled, Sony pictures shot the movie Fools Rush In, which offers this footage of it in action (well, not really in action, and not lit up, but it's still good footage of it!)
The sign is now resting peacefully in the Neon Museum’s Boneyard. You can see it, and plenty of other glorious neon from Vegas’ illuminated past, on a tour of the facility.
