There is another difference to be aware of:
With ASCAP - if you are publishing, you must register as a publisher. This really isn't "self-publishing", because the publishing name you choose should then be registered as a business entity in the state you reside for you to be able to legally assert your right to collect publishing royalties.
With BMI - if you register as a songwriter you immediately become a self-publisher. There is no need to register as a publishing entity also. You can assign publishing rights to yourself when you register your songs with them.
In either case, always file appropriate copyright paperwork with the registrar of copyrights to protect your works, regardless of their published status.