It's unclear if they're actually related at all or it's just one big misunderstanding perpetuated over the years.

https://en.wikipedia.org/wiki/Slavs_%28ethnonym%29

"The Slavic autonym *Slověninъ is usually considered a derivation from slovo "word", originally denoting "people who speak (the same language)," i.e. people who understand each other, in contrast to the Slavic word denoting "foreign people" – němci, meaning "mumbling, murmuring people" (from Slavic *němъ – "mumbling, mute"). The latter word may be the derivation of words to denote German/Germanic people in many later Slavic languages: e.g., Czech Němec, Slovak Nemec, Slovene Nemec, Belarusian, Russian and Bulgarian Немец, Serbian Немац, Serbian, Bosnian and Croatian Nijemac, Polish Niemiec, Ukrainian Німець, etc.,[3] but another theory states that rather these words are derived from the name of the Nemetes tribe,[4][5] which is derived from the Celtic root nemeto-.[6][7]"

"The English word Slav could be derived from the Middle English word sclave, which was borrowed from Medieval Latin sclavus or slavus,[47] itself a borrowing and Byzantine Greek σκλάβος sklábos "slave," which was in turn apparently derived from a misunderstanding of the Slavic autonym (denoting a speaker of their own languages). The Byzantine term Sklavinoi was loaned into Arabic as Saqaliba صقالبة (sing. Saqlabi صقلبي) by medieval Arab historiographers. However, the origin of this word is disputed.[48][49]"

https://en.wiktionary.org/wiki/Slav#Etymology

Commonly thought to derive from Proto-Slavic *slovo (“word”), thus meaning "those who speak meaningfully" and contrasting with *němьcь (“foreigner”, literally “dumb/mute person”). However, that word is an s-stem and thus the inflectional stem of that word is *sloves-, so it cannot be the direct origin as it would lead to an expected form *slovesěni (compare Russian словесность (slovesnost')). Thus the most likely origin is the verb *sluti (“to be known”).[1] Both words ultimately derive from Proto-Indo-European *ḱlew- (“fame”).