Krystal D’Costa accounts for why we sign emails with “Thanks” instead of “Sincerely”:
Email offers a speedier means of contact than an actual letter (and in some cases, a telephone), but that speed also means we’re sending more messages through this medium both for personal and professional reasons, and reading and responding to these messages requires a commitment of time. So it’s more important that the sender recognize the burden that they’ve placed on the recipient.
In a time when letters took time to write, send, and respond to, it was important for the sender to attest to her reliability. Responses and actions were not so easy to take back. “Sincerely” and “Yours truly” which were meant to build trust between communicants. Credibility was an important determinant of whether a response would be issues. Today, as the web enables stranger to contact each other with little effort, credibility is less of a factor in determining responses (SPAM mail aside) when weighed against time.