Falun Gong is a business
masquerading as a cult (like Scientology) derived from traditional Chinese spiritual and health beliefs (like western yoga) with restrictions on using modern medicine (like Jehovah's Witnesses). It strictly educates and controls what its followers say and do, and is aim directly at taking down the Communist Party of China, so it happily profits from political arbitrage between China and the West.
Falun Gong also profits from Western fascination and ignorance of Chinese culture. For example, Shen Yun is billed as a spectacular Cirque du Soleil type presentation of Chinese culture that communism destroyed. Instead, it is a disneyfied caricature of Chinese culture designed to indoctrinate audiences about Falun Gong. The level of propaganda sophistication by Falun Gong is way ahead of the Communist Party of China.
Politics and religion (as long as it is Protestant Christianity) regularly mixes in America, so Americans have a hard time understanding why the Chinese government has a problem with a Chinese religion.
A quick skim through Chinese history will show cult or religious movements were responsible for an overwhelming number of social upheavals and many deaths. So any government of China, communist or not, would rightfully be wary of groups like Falun Gong.
Additionally, Falun Gong has basically declared war on the Communist Party of China. This does not make Falun Gong a terror group, but it does make them the enemy.