Bird records committees are established in most (all?) states and many countries. They are generally not "official" in terms of having any link to the government, but they create "official" lists of the birds seen in the state and attempt to be the final word in the birding world for their areas. They are usually a group of volunteer birders that are well respected in their communities. (In Utah, they are selected by the nomination and voting of other members of the committee only, but the details of their composition and bylaws vary by state and are a separate issue from their role in birding.) Their primary purpose is to evaluate records of rare birds in their area and to judge whether those records should be included in their "official" records. As a result, they compile documentation supporting when and where the rarest species occur in the state, and attempt to provide an indication of whether each of those data points has been established beyond reasonable doubt.
EBird is an online citizen-science tool that is run by the Cornell Laboratory of Ornithology. It began by accepting records from North America but has since expanded to global coverage. Its goal is to use the power of millions of birders around the world to study the abundance and distribution of every species of bird through time. Although it includes rare birds, its strength is in its numbers: it focuses on all species, regardless of how rare or common they are. By doing so, it can provide estimates of relative abundance that are lacking in many other citizen science projects. The data submitted by eBird users are already being used to generate the most detailed distribution maps ever made, and this is only the tip of the iceberg in terms of how such data can be used by scientists.
Bird records committees are concerned with (locally) rare birds only; eBird is concerned with all species. BRCs are concerned with one state only; eBird is concerned with birds of the entire planet. BRCs attempt to create a list of rare sightings; eBird attempts to create a database of all sightings, and to incorporate relative abundances, search effort, and other scientifically valuable data. BRCs create lists for the state; eBird provides data for scientific analyses.
Certainly there is overlap between the coverage of these bodies. When a rare bird is observed, both eBird and the BRC want to hear about it. But the kind of data they seek, and the way they plan to use the data, differ greatly. If you want to contribute to citizen science, and to have your bird observations help biologists understand the distribution of birds in time and space, submit your records to eBird. If you want to help your state maintain an accurate and vetted list of the species seen in your state, submit your rare bird sightings to your BRC. And when you are lucky enough (luck tends to favor those who work hard) to find a rare bird, consider sharing that observation with both.