Neither FTP clients, WebDAV clients, nor web browsers are very well suited to doing cryptography as part of the upload/download process, so accessing your data this way would require the server to first decrypt your data, and then give it to your browser. This breaks our no knowledge model, where SpiderOak does not have your encryption keys, and SpiderOak servers are entirely unable to decrypt, read, or send any data.
So, SpiderOak's approach to uploads and downloads is that upload transactions are created in encrypted form by the local client, then uploaded to the server. Downloads happen in the reverse order. Never does a server know any plain text data -- not even file or folder names. In the datacenter, we store only sequentially numbered data blocks. If you lose your password (from which your encryption keys are generated) we cannot recover it for you.
Nevertheless, convenient web access is important to some people, so we did build a web access option. It stores the user's password (and derived keys) in memory only -- never writing the session file to disk -- and purges them from memory as soon as the session ends. On our website, we actually recommend against using the web interface at all due to this issue, but we provide it anyway, due to overwhelming customer demand.