Bookkeeping involves more of the day to day transactions of an organization. This is essentially keeping track of a client's financial transactions through computer systems. Bookkeeping can be a good start if you are interested in accounting, but accounting involves more than just numbers. It includes interacting with clients, giving advice, and having a bigger view as to what transactions mean for a specific organization.
With the rapid computerization of financial records, I want to add that bookkeeping isn't something organizations are looking for a person to do anymore as they expect their accounting system to perform these duties.