na-adjective
Literal use for things that can explode or are related to explosions, such as 爆発的噴火 (explosive eruption).
火山が爆発的な噴火を起こした。
The volcano caused an explosive eruption.
na-adjective
explosive (figurative); sudden and dramatic
Describes rapid, dramatic increases in sales, popularity, growth, etc. Often used with 増加, 成長, ヒット, 人気.
新商品の売り上げが爆発的に伸びた。
Sales of the new product grew explosively.
その曲は爆発的な人気を集めた。
The song gained explosive popularity.
急速 means 'rapid' or 'high-speed' and is more neutral, often used for steady fast progress rather than a sudden burst.
Compound of 爆発 (explosion) + 的 (-like, -ic). The figurative sense developed from the image of an explosion.