Yes they are separate. dmoz is a directory moderated by volunteers. dmoz listings are apparently very good for seo, possibly because the directory gets syndicated on a lot of other directories. The Yahoo directory is a separate directory managed exclusively by Yahoo. As far as I know the Yahoo directory results are only shown on the various country specific yahoo directories. These listings are good for seo too, but possibly not as effective as they once were.
Maybe , you have got things confused between yahoo and google. As stated in previous messages, DMOZ and YAHOO are separate. However, GOOGLE directory is just a copy of DMOZ. DMOZ is related to AOL while Yahoo is of course part of Yahoo inc. DMOZ is a free directory (but it takes quite a time to get listed and getting listed is not guaranteed) while Yahoo is a paid directory for commercial sites. There is no relation between DMOZ and Yahoo directories at all. GOOGLE directory is a duplicate of DMOZ directory. Both have the same content (category structure and links). Google directory has no submit option, it just uses the DMOZ data (with a short delay). There is just a difference : DMOZ sorts links alphabetically while google sorts them on a PR basis (not the same scale as the google bar PR).
I have heard many great about DMOZ ebcause they have good team and whole process of authorization is provided by people and it is very good.
Yahoo is not part of Dmoz.Were as yahoo directory is more friendly in yahoo search engine,Dmoz is more friendly in Google.Yahoo also have free submission but it takes longer thank Dmoz to be accepted.For me Dmoz is better becasue 95 of my SE referal is from google.
I can add that one of the best advantages of DMOZ is high rating in google Listing in this directory is perfect start for your website, but unfortunately listing takes more than year
I think DMOZ is part of AOL, not Yahoo. You can check on DMOZ site. they put AOL logo on their site and said as AOL partnership. Also, you can check DMOZ using whois domain tools. DMOZ registrant using AOL email.