I'm on the Bing Maps team and have worked closely with HERE for many years.
Bing Maps uses a number of different data providers for different types of data. Most of the road data comes from NAVTEQ (now owned by Nokia HERE) and has been the primary data provider for Bing Maps since 2005 I believe. In certain countries other data providers are used. Most notable are China (AutoNavi), South Korea (SK Planet), Japan (Zenrin). TomTom is used for some data types such as Postal Code boundaries, but I don't believe their road data is used. Ordnance Survey is used for Postal Code data in the UK.
The aerial/satellite imagery comes from a number of different sources. Most of the imagery in the US and Western Europe was captured by the Bing Maps team themselves using camera's that they built themselves (hired companies like Digital Globe to fly Microsoft camera's). The Bing Maps team actual bought a company many years ago called Vexcel who make some of the most widely used camera's for capturing aerial imagery. Digital Globe has been a data provider for Bing Maps for a long time as well. Satellite imagery is also sourced from NASA. Birdseye imagery comes from Pictometry and Blom.
As for comparing Bing Maps to HERE, Bing Maps combines a bunch of premium data sources together including data from HERE to provide a best in class set of data set. HERE does have some benefits in that they do have some older NAVTEQ services that provide some more in-depth functionality that most Bing Maps users don't need, such as Truck based routing. That said, being that HERE is one of the main data providers for Bing Maps, you can license these in-depth services on their own from HERE and use them with Bing Maps. I actually have a number of customers who are doing this as they prefer the imagery, data and services available in Bing Maps which is what they need for majority of their use cases and they also like the enterprise level support that all licensed customer have access to at no additional cost.