While it’s not an exact match by any means, it seems to be quite reminiscent of fast vs quickly in English. Notably, fast and quickly are quite similar, but have distinct definitions and uses.
The „as soon as, immediately after“ meaning for そうそう is often used as a suffix. My dictionary lists examples like 入社早々, 開始早々, 新年早々.
Also, the other meaning for そうそう „quickly, promptly“ is often used in the form of そうそうに (仕事を早々に切り上げる), while はやばや (with the same meaning) is more likely to be followed by と (早々と投票を済ます).
A hint from my 明鏡 dictionary explains this quite clearly: