I couldn't find a mention of Liberty Basic on that site, but assume you mean this one -
http://www.libertybasic.com
Differences between that any Basic language and the Basic used for the PICAXE are numerous and huge. For one, PICAXE Basic is used for programming a PICAXE alone, while Liberty BASIC is a general purpose programming language for Windows PC's.
While they both serve a common goal of allowing people to undertake programming, and both are suited for the job they do, those are perhaps the only things they have in common.