They all make good products. However, LGB has been around the longest and caters to a wide variety of interests, including North American and European outlines. Aristocraft and USA Trains cater mainly to the North American prototypes and they do make very high quality, detailed trains, modeled very closely after the prototypes you see on railroads today.
So for someone like myself, who is interested in European prototypes, LGB is the only game in town. While LGB has now moved most of their production to Hungary (via Marklin) or China, the majority of their products were manufactured and assembled in Germany since 1968. That coupled with LGB's high quality standards have ensured that their products are robust and stand up to all kinds of abuse.
They design their products to be operated in rain and snow, and actually encourage this, since one of their mandates was to allow everyone to enjoy their products no matter their locale or space situation. I have run the trains in snow and rain and they are not any worse for wear since the mechanical parts are well sealed from dirt and moisture. You can leave their trains in the sun for years and the plastics will never fade or crack. The use of advanced plastics means that the detail parts are flexible and don't snap off from even rough handling.
The track they manufacturer, while not exactly prototypical, fares fare better in the elements than any other manufacturer, whose plastic ties tend to deteriorate from the UV rays in sunlight. Not good if you have spent thousands on a garden railway, only to find you need to replace the track a couple seasons later.
LGB is also known as a 'toy' train manufacturer - their starter set like the one Shaygetz posted above has been manufactured in the hundreds of thousands, and many of those early trains are still in use today. While certain trains they have do look 'toy' like it's partly to appeal to youngsters as well - the trains are robust enough to be played with by younger kids, whereas you wouldn't really give a Dash-8 locomotive to a 5 year old. That locomotive above, the Stainz, is actually modeled from a prototype that was used in Austria many years ago. Since it does look funny, it may not appeal to those who are interested in exact scale replicas.
As Shaygetz mentioned, his starter set was one of the first manufactured and still runs like it was brand new. Some of my LGB models are 20, almost 30 years old and they still run well and look the part.
LGB, like many model train companies as of late, did file for bankruptcy a couple years ago and have eventually been bailed out by Marklin who has resumed LGB production this year and there is a product line for 2009. Unfortunately, this does not include any North American models, only Euro at this time.
So, I may be biased a bit in my presentation of LGB, but I am a strong believer in quality products that last.