From a metallurgical and weld integrity standpoint, there is no advantage of either lace or weave.  The best grain structure and weld integrity come from stringer beads.  Weave beads have more heat input, larger grains, and can trap slag which can cause failure of impact tests when certifying or compromise weld integrity in service.

Most weld specifications limit weave for this reason.  However, some grace is required for the weldor who is forced to work in the real world because it is realized that weave may be necessary to fill root gaps and for other reasons.  The specs I work with every day limit weave to 3 x electrode diameter or 10mm, which is less than 1/2 inch.  And, we often restrict critical welds to stringer bead only.

I think Lace is just a big weave that goes side to side and up and down if I am not mistaken.  Not sure what it means in weldor slang, but I think I have seen it.  It looks awful.

I see now your post is regarding GTAW, so the slag comments do not apply - but the grain size and poor toughness are still a concern with wide weaves.

When people talk about weave it is usually SMAW or FCAW and they always mention slag entrapment. That is why I went off half cocked.  LOL