In the original Creed, 'proceeds' (in the Greek, which I can't reproduce here since I suck at exotic characters but the 'romanized' word is ekporeuomenon) means 'originates from and is granted by' with the belief that the Holy Spirit, coequal with the Father and the Son, nonetheless is created by the Father and goes out to the Church Militant (the pilgrim church on Earth, as opposed to the Church Triumphant in heaven and the Church Suffering in purgatory) to aid in the faith, both in maintaining it and in spreading it.
The Latin, however, the word they use for 'proceeds' doesn't include the denotation of originates from, therefore in order to not fuel the Arian heresy, the Creed includes the Filioque 'from the Father and the Son' to emphasize the co-equal divinity of the Trinity. This is not a concern in the Eastern church, because they didn't face the Arian heresy as heavily, being rather more concerned with Monophysite and Monothelite heresies. Plus the Greek term is more extensive, thus doesn't need to specify 'and the Son', indeed adding that to the Greek would simply fuel those controversies that weighed most heavily in the East by seemingly dividing the Father and the Son.
In the end, it all comes down to the language used, in the Latin procedit (proceeds) has no denotation of creation, only of procession (IE being granted to mankind by), whereas the Greek does have said denotation.
Here is the Latin Creed of that section
Et in Spiritum Sanctum, Dominum et vivificantem
qui ex Patre Filioque procedit
Qui cum Patre, et Filio simul adoratur, et cum glorificatur.
It's semantics.
To make things more interesting... the filioque actually originated in the East, not the West, specifically by Cyril of Alexandria, and for several centuries it spread through the West as an anti-Arian measure, was taught by multiple Popes who are considered Saints by both East and West, and was in general circulation. The first instance of objections to it in the East came from Paul II, who inserted it into his side of the dueling excommunications between himself and several Popes (and said objection got slapped down hard by Maximus the Confessor who pointed out the linguistic issues, and also that Paul II was a Monothelite and thus indeed a heretic).
Then it erupted again in the West in the 8th and 9th Century because the Frankish Church never got the memo that the filioque only was part of the Latin creed, not the original creed, and the Carolingian emperors tried to use it as a club against the Byzantines, with the Popes generally calling foul and telling the Carolingians to stop trying to help, because they weren't helping. In fact, between 800 and 1000 there was no Creed sung in Roman Rite masses at all. But the Carolingians and the Franks really liked the filioque, so they kept at it quite cheerfully. Not helping was the Spanish Adoptionism heresy, which the filioque helped suppress.
Then you have Photius, who as part of his attempts to cling to the Patriarchate of Constantinople pretty much started accusing everybody else of being heretics and by virtue of being an extremely eloquent writer and being very highly educated (along with having the backing of the Emperor) pretty much ignited the filioque dumpster fire. You had dueling councils, with Photius at one point pretty much declaring all of the original Church Fathers (Cappadochian, Greek and Latin alike) as heretics because none of them agreed with his extreme monarchial view of the Father. Photius is pretty much the single largest reason that the controversy is still around, he's revered in the Eastern Church as a saint, and condemned in the Western church as a heretic.
So by 1000, the Western church finally went full filioque, and then as the fortunes of the East continued to decline, the West started getting muscular about things 'you want our help? First, follow OUR interpretation of Scripture'. Not particularly nice, but considering that the Eastern Church had been doing the exact same thing for several hundred years at that point when they were in the ascendant, well... understandable.