I have a client who is still using Visual Studio 6 for building production systems. They write multi-threaded systems that use STL and run on mutli-processor machines.
The biggest problem that we've seen at my workplace is it's inability to handle even marginally complex templated classes or functions. This fact alone has force some of the most devoted VS6 fans in the company to upgrade and start using VS2005. In addition to the template problem, intellisense is much better, debugging is easier and more accurate, and many people find the IDE easier to navigate. The only downside that we have seen thus far is that builds take a bit longer in 2005 than they did in 6 (but that's probably a side effect of the compiler being more robust)
You can also check out these sites for a sampling of known issues in VS6:
I'm sure you could find more if you poked around a bit.
Third-part libraries support only a limited number of compilers, too. Your client may not be able to accept bugfixes or feature upgrades as a result.
For instance, even a widely used library as Boost supports only VS 7.1 and later (source)
And you might have some problems with Data Execution Prevention (DEP) as well, because VC6 ships with an old ATL version. As usual, see Raymond Chen for details.
Visual Studio 6 is not compatible with the lasted Windows SDKs, so it cannot utilize (at least easily) the latest OS features.
VS6 does not compile code according to the current C/C++ standard. For example,
Not compatible with Vista. Heck, there's a long list of issues VS 2005 has with Vista.
That being said, most of the improvements in VS seem to apply to everything other than C++ native code. What I'm seeing is more standards compliance, which is important but hardly dramatic.
I have upgraded my stuff but it's relatively uncomplicated. A con to upgrade is VS 2005 DLL Hell