There are three major ones.
The first problem:
InfoPath tries to assist the user when completing forms by popping up little boxes containing messages on the requirements for each field (controls like text boxes etc.).
Unfortunately the browser will more often than not place these boxes over the fields the user is trying to fill in (this is due to the way the MS scripts reacts to the onMouse events). Because a field needs to receive focus before you can fill it in, you either have to tab to the particular field, or you have to click on it. If the popup is in the way however (when using the mouse), the field cannot receive focus and therefore you cannot fill it in. This forces the user to move the mouse around in an attempt to get a clear shot at the field (the popup tries to follow the mouse pointer and sometimes ends up sitting right in front of the field you want to fill in).
Maybe I just haven't played enough with InfoPath, but this is what happens by default when you add rules to the controls. If MS had given more attention to the way they handle events, this could have been avoided. In the meantime I have to space the controls apart to such an extent that the popups won't affect the users.
The second problem:
You cannot use spaces in a multi-line textbox. The form crashes out and you get a really funky error message in your application log: "Exception Message: Reference to undeclared entity 'nbsp'"
How could they have missed this? Their workaround of course can be found here: http://support.microsoft.com/kb/931426
They suggest that you replace all these multi-line text boxes with Rich text boxes. So now a visitor can type in his Postal address in multiple fonds and colors (maybe this can be switched of to some extend but I don't see this as an acceptable solution and the problem should not have existed in the first place).
The third problem:
You can not submit a browser form as an anonymous user if it is configured to save the results into a list. (You can, but you will have to jump through several fiery hoops, and then fly around the moon twice)
As far as I know, MOSS has replaced their CMS product and in that regard they are touting it as one of the best solutions for an Internet facing website (they actually used the words "the best" when they sold it to us) but it cannot even handle this simple task unless you downgrade the form to "email only" (I heard that this hasn't been fixed in SP1)
The "MS Gold partner" that is providing all the development and support for our SPS sites are currently testing SP1 so I don't know if it will resolve problems 1 and 2 (or even the printing problem) but we are hopeful.
Problem 3 is a major concern of ours as we were counting on this type of functionality. The Gold partner told us for more than a year that you could not submit InfoPath browser forms unless you were authenticated. It was only after I started working with the system that I realized we could at least do email submissions.