I'm trying to explore if there is any plausible connection between these words. I find them to be somewhat similar in English, where one might carelessly drop the initial "o".
I believe there is some confusion injected in that curi or curus is a Latinization of the Greek kouros, as opposed to puer. Cor is the Latin root for heart, and distinct from the Greek kour, and becomes cœur and cuore and corazon. The meaning meaning of discordia as "separate-heartedness" is clear (thanks to Joonas for putting it so aptly;) but I'm not convinced it has to be the exclusive meaning of Discordia.
It's particular note that Discordia is a goddess, and what I sometimes think of when hear her name is the "κόρη" as in "Divider Girl Divine" = Strife Goddess. In this conception, Discordia is descriptive of Ἔρις.
Obviously dios is phonetically distinct from dis, and yet there is a relationship between "dis" to "deus"--the Lewis & Short entry on the preceding link lists dis (short vowel) as a dative form of deus, and offers phrases such as "si dis placet" ("if it please the gods") and "dis hominibusque invitis", which uses the alternate meaning ("in spite of every body").
Additionally "Dīs" connotes deity, initially Jupiter, and subsequently the inverse of Jupiter as god of the underworld. Cicero suggests that Dīs derives from dives, per the Plutonian association, and divido shares a meaning with the particle dis in the sense of separation, or apartness. The introduction of a v gives us words meaning divine, such as divus.
I don't think the association of the Dioskouroi with discord is purely linguistic. When I think of brothers in general, and these brothers in particular, wrestling often comes to mind, which is consistent with the conception of twins as representation of the binary. I tend to associate these twins with Eteocles and Polynices, but I need to look into this more deeply.