* It breaks web standards like persistent URLs and the back button (granted, there are some workarounds) and eschews native UI elements.
* It is inaccessible to visually impaired users.
* It is built on closed, proprietary technology, which raises both principled and pragmatic concerns (one pragmatic concern is the poor security framework, which might be ameliorated by more eyeballs on the source code).
* It tends to emphasize superfluous visuals over meaningful content (this isn't a requirement of Flash but it's a common result arising from Flash's dominant motifs).
And finally, a posterior observation:
* An awful lot of Flash applications just suck.