From what I understand, they both have the meaning of going public or becoming public knowledge. Is there a more nuanced difference between the two?
表立つ is almost always used in the adjective form 表立った to modify a noun that follows it, or in its adverbial form 表立って. It means something has an outward sign, and it is often used in a negative sentence.
There has been no visible movement so far.
表沙汰 is usually used in the form 表沙汰にする (not * 表沙汰する) or 表沙汰になる to mean, respectively, someone makes something known to the public and something becomes known to the public. The thing thus made public is usually something negative.
A former employee (has) exposed the incident.