I'm confused.
Business Process Management (BPM) is, as to my mind, a process.
Whereas
Enterprise Application Integration (EAI) is not a process - it is a way of linking systems together.
To me, your question implies that only one can be used. However since one is a process and the other is a technology, I believe they can be used simultaneously.
It could be argued that the first is used by management, and the other is used by developers, but again your question doesn't make sense to me, as it might imply that you are considering changing from being a developer to being a manager (or a manager to a developer), but in that case why compare a management technique with a technology? Why not compare a technology with a technology? Or a process with a process?
Regards, Andrew