Scala, 68 bytes

(c:Seq[String])=>c./:(Seq(""))((a,n)=>a:+n.diff(a./:("")(_+_))).tail

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/: is short-hand for foldLeft operator, a is aggregation, ultimately returns the result we want, n is next element

Un-golfed

def NewLettersPerDay(c: Seq[String]): Seq[String] = {
    c.foldLeft(Seq(""))((agg, next) => {
      val existingLetters = agg.reduce(_+_)
      val newDayLetters = next.diff(existingLetters)
      agg :+ newDayLetters
    }).tail
}