There is a 99% chance that this is caused by a browser extension that you've installed. Frequently the offending extension is Ghostery, but it could be any plugin which blocks content from AddThis.
Wow, thanks Travis. I looked through my hosts file and commented out the lines that I have that route any addthis.com host back to 127.0.0.1. Once I saved the file, the modern version of Garmin Connect loaded just fine. This has been driving me crazy for weeks. It makes me wonder why the functionality of Garmin Connect is so totally reliant on addthis.com. It seems like it should gracefully degrade, or issue some kind of a warning. I would get in trouble for making an interface that just breaks like that.