No, mixcloud doesn't let you download (well not officially - there's at least one website that will let you do it though). Afaik, Mixcloud has an agreement with several of the music licensing/rights groups that allows users to post their mixes up, Mixcloud will cover doing so, but on the strict basis that they cannot be downloaded, only streamed.
Soundcloud on the other hand does not have the same sort of agreement over music licensing/rights, and regards itself as a hosting site really for people uploading music for stream and/or download. Therefore, that makes the uploader completely responsible for what they post and labels, etc. are able to make an infringement claim at any time against the user, rather than Soundcloud.
In addition, Soundcloud put in an automatic system last year or so which will check parts of an uploaded track or mix to see if there is any infringement claim from some of the big boys - EMI, Capitol, etc. I've run into this a few times myself with mixes.