Both have a dependency system, but they are quite different in how they work and resolve dependencies.
Apt/dpkg is also a lot more careful about its treatment of configuration files. Debian packages generally ask questions during preconfigure, build a config file, and then run out of the box. But if you already have a config file it will either be left alone or ask you to resolve the merge.
Only differences that I have noticed and cared about as a home user: Yum lets you do `yum search` but Apt splits that off with `apt-cache` (this is annoying because with yum I can just swap out that 'search' with my line editor). Apt tends to be faster for equivalent operations.
If you use aptitude instead of apt* you get most of the features rolled into one command, and it also provides a curses based interface if you prefer that to the command line but are in a situation where a full GUI is not available to you can use an X based tool. I'm not sure about Ubuntu but it has been standard issue in Debian for some time.
I hope one day there's a command line package manager that works with both DPKG and RPM. Something tells me that will expose the real reason people stick with one over the other; remembering the package manager syntax. I feel no great allegiance to one side or the other, there's nothing of real merit I can differentiate them on, the only real result of the split is package maintainers duplicating work.
I have a set of shell scripts which wrap apt, yum, and macports, so i only have to remember one set of commands. They only cover a small fraction of what those tools can do, but they cover 90% of what i do day to day. They were not at all hard to write.
dpkg -S /some/file # tells you what package provides that file
dpkg -L some-package # lists all the files provided by a package
apt-get search <regex> # lists all package names matching <regex>
There used to be a massive difference. Mostly because debian had apt-get and Redhat had just rpm. That's why we still have sites like rpmfind. Nowadays there is not that big of a difference from end user point of view.
As a dev I still love dpkg-buildpackage way more than rpmbuild.
