Although some HTML apps have this problem too, Flash is invariably laggy.
Flash doesn't work well on Linux, and many of the people in my world run Linux desktops. (And some won't run proprietary software on principle.)
The UI of Flash apps is always some homegrown awfulness that tries hard but is never as intuitive or responsive as native UI, and not even as good as web browser controls.
With Flash apps, printing usually doesn't work as you expect.
You can't save the page.
Flash apps don't really exist on the web. Even if you give each Flash app its own URL, that just means you re-download the same framework over and over again and then load the data.
It is cumbersome (for instance) to bookmark, or bookmark a particular view, or to link directly to other content, or to have a task expand to show an iframe of other content on the web. Some advanced HTML apps have similar difficulties in practice, but Flash apps have these issues in principle.
None of my existing web browser plugins work well with Flash content. For example, some people have spellcheck built into their browsers; now they can't use that when entering data. Even something as simple as cut and paste sometimes doesn't work quite right.
