This very much depends on what tradition you're working in, what information you're trying to convey, and who will be reading it.
As many other answers have noted, the "feel" of the two meters is often thought to be different. This is because 6/8, in traditions that rely on sheet music written in the western style, is nearly universally interpreted as two groupings of three beats with strong emphasis on the first beat and a situational weak emphasis on the 4th. By contrast, 3/4 is interpreted as either one grouping of three beats, or as three groupings of one beat. Playing the 3 against the 2 is what makes 6/8 a compound meter, and when conducted by a human, the pulse is conveyed as 2 beats per measure.
When we start discussing waltzes the distinction becomes more or less muddled depending on your perspective. From a technical standpoint, you could take your copy of "An der schönen blaue Donau", scratch out all of the 3/4 markings, replace them with 6/8 and the strict definition of note-duration would still be accurate. The trouble is that waltzes are not played "straight" even when written in 3/4, and a major part of their tradition is to extend the duration of specific beats depending on the piece.
Very few musicians would be able to sight read your 6/8 waltz, and those who could would find it an annoying way to notate it. With a live orchestra, the conductor may also end up dropping the baton on beats that don't line up with the intended primary pulse of the piece, however this would get sorted after an initial sight-read and the realization that there should be 3 beats per measure instead of 2. A really wonderful rule of thumb for music engraving in general is to make the pulse visible to the musician, and you can see this in the beaming of 8th notes very clearly.
Notably, all of this goes out of the window if you're not working with acoustic instruments or a conducted orchestra. There are certain situations, in pieces of music that have multiple time signature changes and a compound pulse, where it may be appropriate to notate a measure in 6/8 when it has 3 pulses per bar. It ends up being completely contextual. If you're composing electronic music, your equipment may not support 6/8 and it will be easy enough to imply the same feel of stressed and unstressed beats in 3/4.
Writing music and notating music are not equivalent processes, and very often notation falls short of what the composer is trying to convey. When that happens, we must rely upon convention, and in this case the convention is that the two meters have different primary pulse divisions.