Conventional reactors also produce plutonium; in fact, there's a suspicion that one reason Chernobyl was so poorly designed (e.g. lacking a containment dome) was that they were extracting warhead fuel from it. About a third of the energy in US reactors is actually from fissioning plutonium, but they do that inefficiently so plutonium and other transuranics make up most of the long-term radioactivity in high-level nuclear waste.
On the other hand, breeder reactors don't necessarily make weapons-grade material. The Integral Fast Reactor in the US produced a mix of four plutonium isotopes, which was totally unsuitable for weapons, and harder to enrich to weapons-grade than natural uranium.
Using breeders multiplies the energy from uranium ore by a factor of a hundred. With such low fuel consumption, it also makes seawater extraction pretty easy, which extends our fuel supply to many millions of years.