The Bitdefender engine that other vendors are using is usually the same what BD uses itself. I'm not sure if BD offers any variants as not much is known to be able to positively say that.
However, Bitdefender has added Cloud detection in their own antivirus for static detection of malicious files. This is not available to the other vendors using BD engine from what I see.
they can have different detection rates. Why would this be? For example a lot of AV vendors use bitdefenders definitons but they seem to have markedly different detection rates. Now you would think a product like Emsisoft would always be better since they have a very strong BB, and their own definitions on top of bitdefenders definitons but that always don't seem to hold true.
You've kindof answered your own query.
Also note that the definitions are used for static detection (malware are detected before they execute). The BB comes under dynamic protection. Which detection are you considering when comparing engines?
Two things to note are:
First, there may be a time difference between when BD engine is updated for their own antivirus and when the borrowed engines are updated, either due to their contract (how and when are updates made available) or by the borrower vendor's update preferences.
Second, as
@mlnevese said, an AV vendor might be using older version of BD engine causing some difference in static detection. Zonealarm was using an old Kaspersky engine until April's. Only ZA Premium offers Kaspersky's cloud engine.
Qihoo provides BD engine updates only once or twice daily.
To slightly add to the confusion, here's an incomprehensible screenshot from today, where Emsisoft, Fsecure, GData show BD detection while BD itself doesn't show any on VT. Not sure what's the problem .. This ain't the first time I see this.

PS: The engines available on VT may or may not be the same as available in the respective home products.