Is there a difference between the two? Do they perform the same tasks?
I'm a little confused as I'm not too familiar with email servers, but I'm trying my hand at building one primarily to send transactional email from Java and there is very little documentation on Apache James. At the moment I do have it running properly, but I'm running into issues getting DKIM working properly with it and the frustration has made me consider changing.
So...is Postfix a suitable Apache James replacement? Will I need anything other than Postfix to send email from Java?
Last time I tried Apache James was about 5 years ago. That time, Postfix was much, much more mature and stable than Apache James. However, it seems that James was greatly improved since then.
For me, DKIM problem you have, it has nothing to do with the mail server. It's totally a client task to properly sign email before sending with DK/DKIM.
Answering your other question, you will need JavaMail in your classpath (actually mail.jar, smtp.jar, activation.jar) to send emails from Java.