I own the game, but have never played it. I think it would be better than Axis and Allies. Maybe CWO Marc could enlighten us.
I own two copies of TWG (the game board and the sculpts are both of good quality, which is why I bought an extra copy), but I've never actually played it. As maverick_76 writes, however, the two games are indeed similar, and an A&A player should have no difficulty making the jump to TWG. Both games are powered by the same basic four-stroke engine that can be traced all the way back to the original Risk game: income buys units, units fight battles, battles win territory and territory generates income. In terms of equipment, TWG and A&A both use a variety of plastic military sculpts, and are played on a map which uses an area-movement system, which gives a printed value number to each territory and which show the location of certain major cities.
One notable thing about TWG is that, in addition to the basic rules (the second edition of which was issued just a few days ago), there are lots of supplementary modules available to cover extra elements like diplomatic relations and air supremacy. This aspect should particularly appeal to A&A players who like to include lots of detail in their gaming. You can find the rules and the extra modules here:http://www.thewargame.com/fieldmanual.htm
Another thing which I like about TWG is that the board shows something close to a 1939 configuration rather than the more traditional circa-1942 territorial layout of A&A. One way this shows up is in the fact that, as idk_iam_swiss noted, France appears as France rather than as some sort of Occupied France / Vichy France combination. The board isn't a pure 1939 configuration -- for example, it shows in light green the territories (like Mexico) which will side with the U.S. once it enters the war in 1941 -- but it's fairly close to a start-of-the-war set-up for players who are looking for that sort of map. And the large size of the game board (it comes in three sections, each of which folds into three parts) is another nice element.
By the way, I'd just like to note that the pieces you see on the board in the picture I posted don't show a real game of TWG in progress. The pieces in the photo are for the most part A&A sculpts, supplemented with customized roundels and with other kinds of markers (wooden blocks and plastic Othello/Reversi playing pieces). At the time the picture was taken, I was just experimenting with ways of using different kinds of pieces to represent certain types of information on a game board. An actual game of TWG would look different.