To clarify: dist-upgrade upgrades everything, including installing or removing new dependencies. Upgrade upgrades the installed packages, but doesn't add or remove new packages. In theory only using "upgrade" is supposed to be safer (eg more unlikely to break something) but I have not encountered any problems when using dist-upgrade (disclaimer: when you rely on the standard distro repositories; if you use unstable programs/repos or 3rd party repos/PPAs in Ubuntu, it is more likely that dist-upgrade breaks something).
But I do not agree with what ksdkkdddd said. There is not really a use to do upgrade and then dist-upgrade right after it, because upgrade doesn't magically fix something in case dist-upgrade breaks something. It works or it does not. Also if you use apt from the terminal (as you should) and not rely on graphical updaters you always get output on what may prevent upgrading and how to resolve it.
In my experience a package's ability to cope with a complex uninstall scenario generally improves with the initial upgrade - or any upgrade for that matter - (for example, sometimes dist-upgrade-specific issues are only discovered by the maintainer as part of the coordination for a release) and so (in my experience) the reliability of the "dist-upgrade" is usually increased by doing an "upgrade" first. Dist-upgrade is one of those things that "should work" of it's own accord, but there are ways of increasing the odds; I have found a pre-"dist-upgrade" upgrade is one. (Sometimes I'm impatient and only upgrade apt/aptitude before a dist-upgrade though.)
I would say it makes more of a difference on systems that are infrequently upgraded in general since a few points away from the current versions aren't going to contain many differences anyways, but I'd certainly be more hesitant to dist-upgrade from a .0 release (6.0 to 7.x, for example). I'd almost rather clean-install in that scenario since it amounts to a similarly-sized download.
