Demos were ment to be shared freely with people in the hope that said individual would be so interested in the content that they would purchase the final game even Money Grubbers like Bethesda allow demos of some of their games to be distributed.

Correct me if I'm wrong but isn't Beta a pre-release mass test of a game? once the game goes live there is no such thing as a beta version, just the final version
Beta State Games
Beta state games are in some cases a simple fine tuning of a 90% complete game, or could be another stage for the devs to add another round of content for the end product itself. There are a lot of different ways to see what beta state games are and normally they are categorized by the different numbers associated with them. a common way of seeing this (in most games but not all) are the 3 numbers in patch notes such as 0.1.9 (early beta phase) or 0.9.8 (late beta stage) different developers have different ways of doing this but the higher the number, the later in the creation it is.

Now this is a time when its like going over a game with a fine tooth comb to find all the last existing bugs or balance changes. At this point more and more players will most likely take a stance on if the game is a good one or not as most of the time (not all) the future potential has been revealed and what the devs work ethic is and how it will be possibly during later development and release. But this stage is critical as well, because sometimes large bugs, exploits, glitches and other issues do pass over from the alpha state. But its also the time when a lot of new content can be added, tested and looked over.

At this point in development keep an open mind, but keep active on the forums, look for issues and things that might need to be addressed by the devs. Or bring up new ideas and suggestions on in game stuff and mechanics

Review
-Less bugs and issues
-Better balancing
-quicker hotfixes or updates
-more content (but not all)
-game changes

Some developers allow you as a beta tester to continue to use their product for a said period of time, but if youy want DLC's or added content your going to have ot pay like the rest of the fanbase.