I think when 'new writing' is referred to, the terminology is very specific - it is meant to designate a piece that has been written by an unknown playwright, as opposed to one who is well-known and currently writing. It can be used in a widely ranging vein - 'new writing' may be the very highest newly commissioned work in the country at some of the most prestigious venues from up-and-coming scripters; at fringe levels, it can be a bit of a euphemism for what is, in essence, an uncommissioned play, put on for the sake of promoting a company...but the term is used loosely.
'Modern' playwriting was, once, new writing, I suppose, back in its day, but the expectation, when 'modern' audition speeches are asked for, is that they will derive from published plays, and be (by and large) attributable to known playwrights, who have, in effect, become part of a canon of recognised exceptionalism. It's not a hard and fast rule, and some actors swear by working from their own written audition material or using work they have taken from uncommissioned pieces in the audition room and calling them their 'modern' piece (after all, the words are, indeed, contemporary) - but, generally, it's assumed that the essential quality of writing is what has garnered a play a recognised place in the repetoire (and, indeed, been good enough to ensure publication) - and that better writing makes for better speeches, therefore, hopefully, better performances. 'Modern' tends to be used in contrast to 'Classical' in what is again, both a vague and specific sense. It's vague because, as I pointed out a post or two ago, quite where you define the cutoff between 'modern' and 'classical' is very arbitrary and depends on whether you feel this is defined by the inception of a consciously 'modernist' movement in theatre (and, indeed, whether that occurred in the 19th century, the 1910's, 1920's or whatever), whether you think it's defined by a turn to greater social naturalism that came in in the 1950's or whatever. But it is also specific because, in terms of canon, a 'Classical' work is enshrined as a recognised piece of lasting theatre that preceded the 19th century period, and was written in what is recognisably a 'Classical' mode of writing; even Shakespeare, who actually held very little respect for classical rules of structure, still wrote in an essentially classic mode. Before you can get a 'modern' school of writing, there has to be a consciously 'modernising' movement taking place in terms of what is acceptably written about (say, representing the lives of the middle and working classes, rather than the aristocracy), what structure is acceptable for language to take (prose replacing verse being the most obvious shift) and so on. This is certainly not a conscious distinction made by writers before the very late 18th century at the earliest, which is why, in a sense, all 'modern' writing was once 'new writing'.
