To begin, 拂菻 was likely not pronounced "fu-lin" back in the Tang Dynasty. "Fu-lin" is the more modern Mandarin pronunciation that became simplified, standardized, and popularized in the 20th century. The spoken Chinese before that, often referred to as 漢語, has more characteristics preserved in Cantonese and Hokkien/Taiwanese. This means that 拂菻 would sound more like "bo-dim", which is likely just a shortened "Byzantium" or "po-rim". Other names include 普嵐 [pu-lan in Mandarin] (from 魏書·高宗紀 Book of Wei), 伏盧尼 [fu-ru-ni] (from 北史·西域傳 Northern History), 拂懍 [fo-lin] (大唐西域記 Great Tang Records on the Western Regions). These are believed to be derived from Arabic/Iranian/Parthian/Persian names for Rome - Frwm, Purum, Hrom/Hrum.

The name 大秦 came about in a different way. Instead of borrowing the pronunciation, the name 大秦 was given due to how prosperous the Chinese viewed Rome, ranking the Roman civilization on par with their own Qin (秦) Dynasty.

In 後漢書·西域傳 (Book of the Later Han: Tales of the Western Regions) visits and findings of Roman society is recorded, giving much praise to its people, the ruling system, merchandise, treasures, etc... including this line "其人民皆長大平正，有類中國，故謂之大秦" "Its citizens are all righteous, much like China, and therefore referred to as Great Qin"

The name are most likely interchangeable when referring to the general region. With 拂菻 being a more modern (at the time) direct translation and 大秦 more of a classical honorific.

Edits

As Pytheas brought to attention below, the Byzantines didn't refer to themselves as such. I'm not as familiar with western history and made an educated guess at first. I've updated with some research backing above. Thank you!