What is the Postmodernist criticism of history, and why is it significant to the profession?

This is an enormous question to answer. I would recommend that you read the brief, and very instructive, Postmodernism for Historians by Callum Brown.... an enthusiast, admittedly, but one who carefully lays out his arguments. (Of course, you might also look at Jean-François Lyotard's Postmodernism explained to children, just for fun).

I think a good answer would start, however, with the separation of "event" and "text". We can accept that events happen in history -- from large to small. However, any attempt to describe these events is not a "fact", nor is it the "event" itself. Rather it is a "text" -- a non-real representation of event. Books, movies, paintings.... they are all texts.

We can look to Michel-Rolph Trouillot for help here, in silencing the past. Trouillot helps point to the many ways in which "text" doesn't equal "event". Texts cannot capture the complexities of events. Texts make use of only a few of many potential perspectives on what "actually happened". Perhaps most importantly, texts are tools of power, and the construction of history is an act in which power operates. Certain individual's perspectives tend to be recorded as "primary sources", and go into "archives" (in recent history, for example, white, adult, wealthy males... although this is overly simplistic). Historians from wealthy societies, and in recent history from only certain perspectives, then use these sources and rearrange them to tell stories. These stories, the postmodern critique goes, really reflect the historians' milieu and the contemporary discourse in the era in which he/she is writing rather than "event". Thus we cannot make "progress" in writing better histories than those that came before us. (Cue the despair)

The critique goes on then to suggest that the very epistemology of history -- the profession, the rules of peer-review, the format of the 'book' and 'scholarly article', etc -- tends to alienate the "text" (the produced history) from the "event" (what happened). It thus provincializes professional History as one way in which the past is constructed, not necessarily better than heritage production, tradition, nostalgia, etc.

There are a number of good rejoinders to "postmodernism" by historians. I actually don't like Evans' screed. It's angry and fails to see the value of critiques, even ones that we reject. I much prefer Epstein, Barbara, “Postmodernism and the Left”, New Politics, 6 (1997). It's well historicized in a moment and well considered. I also like the more oblique Michael Shermer & Alex Grobman, Denying History: Who Says the Holocaust Never Happened and Why Do They Say It?. This may not initially read as a rejection of core tenets of postmodernism, but it helps to set out some very interesting arguments for what constitutes better or worse histories, and thus give us a sense of why historians can come closer to reconstructing pasts over time.

Personally, I think postmodernism as a critique has added great value to our discipline. I think we've passed the stage where it's an outsider critique or an ideology for contesting sides. I hardly read any scholarly monographs these days that don't somehow benefit from postmodern methodologies or theoretical framings, but I also don't see many award-winning authors (for example) that don't comprehend that they are constructing a past even in the act of reconstruction (i.e. the act of reflexivity).