What are the differences between radar flight following and traffic advisories?
Depends on the ANSP (Air Navigation Service Provider) to what each term means to them. In most ANSPs they mean exactly the same thing or their individual regulations only have one of these terms/procedures available. 

A traffic advisory technically doesn't always require a transponder especially if it is of a snapshot of traffic. The pilot requesting the traffic advisory gives position and a one time traffic picture is provided to the pilot. This is mainly relevant in areas where there is not Secondary Surveillance coverage and/or no surveillance coverage at all. 

However Flight Following does require being able to be identified for continual coverage. Some ANSPs also provide additional reporting services with those receiving flight following like SAR services and Hazard Alerting services.

Some ANSPs only provide these services at the discretion of the controller based on their own perceived workload.

Check the local regulations for the ANSP your flying in to get exactly what you will be provided