In typical usage, the terms seem to be used interchangeably. However, aircraft must be transponder- or encoder-equipped for traffic advisories to be possible, for example.

What information does each provide to pilots that the other does not?

• You can get both by asking ATC for either one. I understand traffic advisories to be one of the benefits of flight following. See this AOPA article on the subject. – J Walters Feb 20 '16 at 19:13
• See also the Pilot/Controller Glossary entries on RADAR ADVISORY, FLIGHT FOLLOWING, and TRAFFIC ADVISORIES. – J Walters Feb 20 '16 at 19:28
• Services provided by ATC depend on the airspace class. When only advisory service is provided, ATC doesn't provide vectors or assign flight levels, only traffic information. This may vary by countries. Typically this is the F airspace (not used in the US). – mins Feb 21 '16 at 14:19