I've had a slightly different interpretation of the differences. I'd classify software into five categories, Freeware, Donationware, Shareware, Trialware and Proprietary.
As Mark said, the definitive rules for the use and distribution of any particular piece of software can be found in the End User License Agreement (EULA), which is legally enforceable. There are also generic distribution rules, which some developers prefer to use. Perhaps the best known for Freeware, for instance are the GPL or GNU licenses, which are used for Linux distributions, for example.
My interpretations:
Freeware
Software products produced by developers who donate them freely to the community, without charge. Usually, these can be copied and passed on to others but in most cases the EULA specifies that the product must be distributed complete, unmodified and including the EULA. For the GNU/GPL licenses, the product may be modified or enhanced but any such enhancements must be freely donated back to the community. If you find a product particulary useful, many developers are happy to accept a voluntary donation.
Donationware
Essentially, the same as Freeware but where the developer asks for a (usually voluntary) donation of some sort, if you decide to keep using the software. Donations requested can be almost anything, from postcards from far flung places to equipment you have no further use for or, of course, unspecified cash.
Shareware
Software products that are developed and distributed by the developer for trial purposes but for which there is a formally defined payment required, should you decide to keep using the programs. Shareware programs can usually be further copied and passed on, subject to being complete, unmodified and with the EULA. Anyone to whom a copy is given is subject to the same "Try and Buy or Delete" rules.
Trialware
I won't get into the argument about crapware on new PCs but that is an example of trialware. Usually proprietary products that are distributed with a specific free trial period, after which, the product will stop working unless you purchase a license for it. A common example is the 60 day trial version of Microsoft Office that is installed on many new PCs. Fully functional for 60 days, then if you want to keep it, you buy a license for it.
Proprietary
Software that is developed and distributed by its developers or their agents, subject to payment of a license fee. Such products must not be copied or passed on to others. Most software from the major producers is distributed under these term and conditions.
Hope that helps.