The user interface flexible and customizable--features are available to please the most visual users, but they can easily be closed to suit those of us who are distracted by flashy GUIs and just want to interact with the database.
Menus and features are not weighted by what Intuit thinks the user will want to see, as they are in QBO. This means that it is easier in QB Desktop to find the functionality that you need. It also makes QB Desktop a better tool for beginners, and for teaching beginners.
The Accountant Tools (especially bulk editing of transactions) are great, and more fully featured than they are in QBO. There are a few tweaks that I would make, but I really appreciate QB Desktop's Accountant Tools.
QB Desktop makes it generally easy to tag data on multiple dimensions, which makes it flexible--suitable for accounting and reporting needs from very simple to quite complicated.
If your tagging needs are very complicated, you may be better off doing data entry outside the program and importing it. Luckily, QB Desktop does not limit the kinds of transactions you can import, and the data's underlying mark-up (IIF) is publicly well documented, open to all users, and fairly easy to learn and use. (All of these are advantages over QBO.)
QB Desktop's Advanced Search feature is one of my favorite things. It allows you to search on dozens of parameters without first narrowing what kind of transaction you wish to search. The flexibility of such a search is obviously valuable, and it also supports multiple use styles; you don't have to think about how another user might have entered a transaction, or who that user might have been. You just enter your parameter, and the database turns up everything that fits. (For this reason, plus a few other minor restrictions that QBO imposes on searches, the search capabilities of QB Desktop are clearly superior to those of QBO.)
The program is rarely slow to respond or laggy.
Multiuser access is costly. Online access requires hosting on an application server, which can be expensive; it can also be complicated, which is a turn-off to unsupported/casual users.
The bank-transaction import/matching feature is a little lacking; it doesn't learn rules quickly and isn't very flexible. If you're quick at data entry and/or your data tagging needs are complicated, you may just decide never to use it.
Repairing bank reconciliations that have been thrown off by changes or deletions is more difficult than it is in QBO. Would love to see the QBO feature that shows you what went wrong and offers you a way to fix it without undoing and redoing everything implemented here.
The audit trail is a little less functional in QB Desktop than it is in QBO.
Lacks a real inventory (costing) system, though third-party solutions are available.
Lacks the ability to store percentages by which costs need to be allocated; this means that you're better off doing complicated allocation data entry outside the program and then importing it; it also hobbles the program for allocated budgeting and forecasting.
Lacks safeguards against importing the same data multiple times.
From a database perspective, the Name field is poorly defined, which can be a problem if you need to report on multiple dimensions (I've run into this problem particularly with non-profits that need very detailed reporting).
When QB Desktop crashes, it crashes hard, but support is generally very good.
QB Desktop suits a wide range of bookkeeping needs, from simple to relatively complicated, and it is very flexible and customizable.
It also allows you to implement DIY solutions, especially to data tagging, manipulation, and import needs, without the need to rely on (and pay) third-party vendors.
If you have little to no bookkeeping knowledge, or are training/working with someone who lacks that knowledge, I far prefer QB Desktop to QBO.
QB Desktop is worth the trouble and cost of licensing multiple users and using an application server for online access.
QB Desktop is an all-around solid piece of accounting software.
Part of my business is teaching clients to work on and understand their own books. QB Desktop is great for this, especially when compared to QBO. (QBO's design and wording favor cash accounting, which is not appropriate for most businesses, but which is intuitive to those with no bookkeeping knowledge; this makes it a difficult tool to use to teach beginners.)