They are slightly different - the ETag does not have any information that the client can use to determine whether or not to make a request for that file again in the future. If ETag is all it has, it will always have to make a request. However, when the server reads the ETag from the client request, the server can then determine whether to send the file (HTTP 200) or tell the client to just use their local copy (HTTP 304). An ETag is basically just a checksum for a file that semantically changes when the content of the file changes.
The Expires header is used by the client (and proxies/caches) to determine whether or not it even needs to make a request to the server at all. The closer you are to the Expires date, the more likely it is the client (or proxy) will make an HTTP request for that file from the server.
So really what you want to do is use BOTH headers - set the Expires header to a reasonable value based on how often the content changes. Then configure ETags to be sent so that when clients DO send a request to the server, it can more easily determine whether or not to send the file back.
One last note about ETag - if you are using a load-balanced server setup with multiple machines running Apache you will probably want to turn off ETag generation. This is because inodes are used as part of the ETag hash algorithm which will be different between the servers. You can configure Apache to not use inodes as part of the calculation but then you'd want to make sure the timestamps on the files are exactly the same, to ensure the same ETag gets generated for all servers.