Problem #1: It's not a PC
The obvious weakness of a Chromebook, of course, is the fact that it's NOT a Windows PC or Macintosh, whatever it is that you depend on to run your specialized application programs. In reality, most of us also have some kind of "heavy" app that we need anyway - Photoshop, CAD tools, etc. These would not run adequately on a machine of the caliber of a low-end Chromebook. There's really no solution for this but to recognize the limitations of the Chromebook and use the appropriate tool when necessary.
On the other hand, adding a Chromebook to your desktop work setup is a cheap way to get a second screen simultaneously available to you for email, referring to a document, and web browsing. Not only does this de-clutter your PC's screen, but also frees up CPU and RAM from your main PC. The additional benefit of decreasing the virus risk to your main PC is worth noting too.
Problem #2: Printing
It's not currently possible to print to a directly attached printer. The Chromebook-preferred method is to use Google's cloud printing solution, which can be a PC sharing its printer with appropriate Google printer software installed, or directly to a Google Cloud Print enabled printer. In either case, the print job has to go first to a Google server, and then down to the destination printer. This seems wasteful of bandwidth, besides bringing up questions of privacy and depends on having a reliable internet connection.
Alternatively, you could simply "print" to a PDF file on the Chromebook and transfer it to a PC with a printer using a USB thumb drive. Or you could use Dropbox or Google Drive to share the PDF printout with a PC with the printer, but that again introduces the same problems as using Google's Cloud Print system -- you are depending on an external server and a working internet connection.
Problem #3: Connectivity to Other PC's
For a Chromebook to be really useful in an office environment, it needs better connectivity solutions over a LAN but without needing an internet connection. Currently Chromebooks can't see the shared files on a LAN using the common Windows or Mac file-sharing protocols, and there doesn't even seem to be a good FTP-type solution, one of the most basic ways of transferring files between computers. Note: It's possible to transfer a file FROM a LAN server using FTP, but not transfer from the Chromebook TO a LAN server.
Again, the workable, but awkward, interim solution is to carry a USB drive from the Chromebook over to a PC, or to use an indirect network transfer method requiring a server out on the internet (e.g. Google Drive).
Hope for the Future
The problems listed above are fortunately "only" software problems, so there is some hope that updates to the Chrome OS software or extensions in the Chrome Store will cure these shortcomings in the future. I hope Google is listening...