Is it a heresy because the Holy Spirit only proceeds from the Father, or because of the way “and the Son” was added?
It is a heresy not mainly because of the Clause itself, which
can be understood in an Orthodox sense (although it remains misleading and divisive) but because of the definitions of the Roman councils of Lateran IV, Lyons II and Florence to defend it. These councils “defined” that the Spirit proceeds eternally and (worse)
equally from the Father and the Son, acting together as one principle. This creates
two principles of origin in the Trinity, the Father and the Father-Son. It comes very close to creating a fourth person, again, the Father-Son. In the patristic and Orthodox view of the Trinity, the Persons are never divided up this way, and are never confused this way. Two Persons of the Trinity do
not act in isolation from a third. The Father is the unique source of both the Son and the Spirit, and, other than that, what is said concerning the Trinity must either be said of the
entire Trinity together, or of one of the Persons, never of two of them.
It really comes down to taking the words of Christ in John 15:26, where He draws a distinction between the eternal procession of the Spirit (“from the Father”) and the sending of the Son from the Father
and the Son, seriously.