What are Minecraft Alpha and Beta and what differs between them?

Solution 1:

Before Minecraft 1.0 there was Minecraft Beta: "The Beta development phase started on December 20, 2010 and ended on November 18, 2011. The main goals of this development stage were a larger focus on polishing gameplay, additional content, and stability. The game's price also increased from what it was in Alpha, but users who previously bought the game were not charged extra."

Before Minecraft Beta there was Minecraft Alpha: "The Alpha development phase started on June 29, 2010 and ended on December 20th, 2010. This version also limited the gamemode to survival, but saw a re-release of multiplayer. At this time, Notch began developing the game full time, causing much more frequent updates. These were also the first versions which had to be purchased, though the price increased in later stages."

So basically, Minecraft Alpha was the first version of Minecraft, when was heavy developing. Minecraft Beta came after Alpha, with the objective of polishing the game before Minecraft 1.0.

Minecraft Versions

Solution 2:

Alpha is older than Beta, has simpler rules, lighter calculating, simpler biome and chunk rendering, fewer items, more bugs, less-worked textures and such stuff than Beta (this is not against Alpha!). Beta has more complicated controls, and working Multiplayer, etc. Beta was also more expensive than Alpha.

Alpha

Starting version: 1.0.0 (June 28, 2010)
Latest version: 1.2.6 (December 3, 2010)
Screenshot:

Beta

Starting version: 1.0 (December 20, 2010)
Latest version: Minecraft 1.0.0 "Full version" (November 13, 2011)
Screenshot: