It depends on the e-mail client, and any post-processing rules the user has set up.
From the most general point of view, there is little difference in that the user receives the message either way (assuming everything is working the way it is supposed to, but if it's not working it would fail in either case the same way).
But if you want to pay attention to details, then they're not identical. The differences, or whether there are any differences, will vary from e-mail client to e-mail client. To give just one example, Lotus Notes will display a different icon in your Inbox depending on whether the message is addressed to you or you are receiving a carbon copy of it. This is a pretty trivial difference, which you may not care about. I mention it, however, because now matter how trivial, it is a difference in how one particular e-mail client handles the presentation of the mail.
More significant are differences based on e-mail rules that a user has set up. For instance, I've often created rules to divert messages from certain addresses to a folder I can ignore, unless I'm listed in the To: field. This is useful for junk mail that comes from within one's company. There are always those who think the whole company needs to know everything they're doing. But I want any e-mail from them that comes directly to me, especially if they're in management.
Differences based on mail rules will obviously vary from user to user, and again: you may not care about them. But if you want a thorough understanding of possible effects of putting an e-mail address in the To: or Cc: field, this is something you may want to consider.
You asked about the effects your choice of field will have on the e-mail client, but the biggest effect is the one it will have on the user. By putting a user's address in To:, you're communicating that the content of the message is directed to them, while by putting it in Cc: you're communicating that it's just an FYI. This affects what "you" means within the context of the message. If a message comes to me that says "you need to do such-and-such" then I take that to mean I have to do it. If it's addressed to someone else and I'm Cc:ed on the message, then I take that to mean the other person has to do it.
You didn't ask about the difference in meaning between To: and Cc:, but that difference in meaning is the underlying reason for any differences in the way an e-mail client handles the message, as subtle or negligible as those differences may be.