Good answer from Midnight. Sex addiction is the psychological need for sex, and can be with only one person. It is often referred to as nymphomania, and that is where the confusion can lie. The connotations of nymphomania are that the sufferer will have sex with anyone, anytime, and that leads people to believe that sufferers are promiscuous. This is not necessarily the case. The need for sex is not a need to have many sexual partners. Promiscuity, of course, is the act of having more than one sexual partner. The proponent may not have sex often, but will normally enjoy the new experiences brought by each new lover. I say 'normally' because sometimes the reason for promiscuity is low self-esteem, and the individual hates themselves while still committing these acts.

Venn diagram time. The two overlap, but are not mutually exclusive.