What are the differences between Jacobson's "Basic Algebra" and "Lectures in Abstract Algebra"?

The Basic texts are much more modern in content and organization than the Lectures. The Lectures are based on Jacobson's graduate lectures on algebra at John Hopkins and Yale in the 1940's and early 1950's-consequently, the style is far more classical and categorical/homological methods are nearly completely missing. Basic also covers quite a bit more than the Lectures. That being said, the Lectures are very careful and comprehensive and it's interesting to compare the 2 via the state of the field in the different time frames if you can get a copy relatively cheap.

Rather than directly answer your question (which I don't have time to do now, and others are probably more qualified anyway), I thought I would mention that you might be able to get some useful insight by looking at book reviews of the later-published Jacobson texts. Here are three reviews that I know of:

Darrell Eugene Haile, Mathematical Intelligencer 3 #4 (1981), 188-189. doi:10.1007/BF03022984

Kenneth Paul Bogart, American Mathematical Monthly 92 #10 (December 1985), 743-745. jstor

Andy Roy Magid, American Mathematical Monthly 93 #8 (October 1986), 665-667. jstor

I second Mathemagician1234's response. Lectures in Abs. Alg. were one of those wonderful old-school treatments in the spirit of Herstein's Topics in Algebra - elegantly readable and lacking in more modern tools like categories and functors. For those of us who've had trouble crossing that bridge, that can be a strength. On the other hand, Basic Alg. doesn't seem to use much in the way of homological methods until Vol. 2, and a glance there suggests that his introduction there is relatively chatty, example-driven, and user-friendly as usual.