Salve Garrule,
Some Romans hated the Greeks because they actually felt a culturally inferior to them, even though they had conquered them. Originally, as someone once said, Romans were military farmers. They were good organisers, planners, politicians, soldiers, merchants and generals but most ancient philosophers, artists and writers are Greek, and if they are Latin they have Greek examples or counterparts. Most Latin writers or members of the high society in Rome were also bilingual. Speaking Greek was considered to be a sign of class, like, for a long time and still to some extent today, speaking French was in the UK (or the US).
So the reason why some Romans so fiercely opposed the Greek was born out of frustration. Part of it also had to do with the age-old cliché that if you are unfamiliar with something, you'll be predisposed to dislike or dismiss it, especially when it begins to pervade your own culture en masse (a lot of Greeks came to Roman cities to look for work, or simply captured as slaves).
There were also those who really liked the Greek and Greek culture and were organised in circles of philhellenes. The anti-hellenism gradually faded as the empire became increasingly multi-ethnic and it appeared that Latin (culture) had not been undermined by Greek influence* at all, but rather strengthened.
Vale bene,
Draco
* Some also considered any and all influence from the East to be effiminate. Funny to remark is that the Greeks thought the same: to them the Persians were a bit effiminate, and probably to the Persians the Indians were weak as well
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Gn. Dionysius Draco Invictus