Much of Android is open to tinkerers, but Google has gradually closed off more and more of the default functionality. The most awesome aspects of the KitKat dialer - its ability to search for businesses and contacts from within the app - were not included in the open source version. So what's a ROM developer to do? Why, create their own alternative. The OmniROM folks have previously shown off their work, and now the CyanogenMod team has packed similar functionality, albeit seemingly more powerful, into the latest nightlies.
Google has been increasingly transitioning new features in Android to a more closed model. Whether you're talking about music playback, search, or even the dialer, Google's updated apps have features not included in the Android Open Source Project. The developers of OmniROM are looking to make the handy features of the new Google Dialer available without the proprietary bits, so they're working on an integrated phone number directory without Google.
Of all the announced new features of the Nexus 5 and Android 4.4/KitKat, one of the cooler ones was a new dialer colloquially known as the "KitKat dialer." It features a revamped UI and includes the ability to search for businesses, contacts, etc. right from inside the dialer. It also features Internet-based caller ID. The name "KitKat dialer," however, is not entirely accurate.
Google Dialer (its official name) is not technically part of KitKat because it is not included in AOSP.