It is harder to answer this question since they aren't in the same category, one is a clip site and one is a file host!
Manyvids is a clip site, which means that people will go there to look for videos to purchase and will buy them through the site. This is awesome because it means (mostly) passive income for you! I definitely suggest that girls who sell videos and are comfortable with it use clip sites like Manyvids and Clips4Sale. You can generate links that you can give to people and they can get the video without having to buy it, so that can be nice in some situations. If your intent is to sell videos on MFC (or some other site) and then give them a download link people might be a bit upset or feel put out by the fact that they have to sign up for an account on some other website to get the video that they just bought. When I heard that Manyvids was going to let us generate links we could give out to downloads I was ecstatic because I thought it would mean I could close my Mediafire account, however the members who tested it for me didn't like having to sign up on an external site and I think that they felt a bit like it was shady. For some people this will be awesome though because they'll have a place they can just log in that will keep their videos for them.
Mediafire is a file host, which means that you can upload pictures, videos, documents, or really any files there. Many girls use file hosts like Mediafire, Dropbox, or the newer Project Maenad for their videos. With these, you give someone a link and then they just download the file. Mediafire doesn't have a password system anymore, I think you can find ways to add them with Dropbox and Project Maenad BUT you can add a password to any file locally by zipping it and password protecting it. In that case, someone could download a file but would not be able to unzip/open it.
We already have a TON of threads about which file hosts to use for videos, people ask this question a lot, but it's really already been answered a bunch of times.
Here is a post with some info and tips about searching on the forum! That way you can find the other threads about this and read about other experiences and opinions!
As for videos being put up on tube sites, torrent sites, etc. it is something that MOST camgirls will have happen to them! Every camgirl should expect that to be a part of the job. It's really up to you whether you want them taken down or not. If you do, you can have your site send a DMCA (don't ever send one yourself) for you to try to get it down. I usually count it as free publicity as long as it has my name on it somewhere. I especially don't mind videos of capped shows, but will sometimes send a DMCA to get content down if it's a video I sell. It's not something I usually stress about too much or dwell on, but I know it can be very different for each model.
As far as security, I would think that Manyvids would be more secure, but I feel really weird about giving that answer since it's not quite a fair comparison since they aren't the same type of site. It's definitely a "comparing apples to oranges" kind of situation. Manyvids is run by someone else, whereas when we use file hosts, we are running things for ourselves. A Target is always going to be able to protect its products better than a person running a yard sale.