Having traveled from Brazil to the U.S, the biggest difference in training I've observed so far has to be the (largely) American preference forgoing any training with the gi in favor of no-gi grappling.
Throughout my own experience, I've come across millions of different reasons why a person should or shouldn't train with a gi. Examples:
"The Gi slows down the game and forces you to think.."
"The gi is a reliable training tool that helps simulate real clothing in a street fight"
"The gi makes you more technical because it just does.."
"The gi is a pointless training tool because it is too unrealistic"
"The training with a gi is pointless because it is too complex for grappling"
"Training with a gi does not prepare you for a fight because you rely too much on grips that don't exist in real life"
The list could go on and on. While the gi may or may not be these things, the thing to understand is that the gi is a form of training tool.
It can be easy to understand why people will often say that the gi is unrealistic. Walk into any IBJJF tournament and sit down for 5 minutes. You will probably see 100 different sweeps and variations of basic submissions that involve wrapping and twisting the gi in all sorts of directions and form your own conclusion that the gi is only a game based on grappling. I believe this is different because as a sport, if something is not illegal then there is no limit to any innovations that will further the sport's competitiveness.
Going back, simply saying that training with a gi makes you a better martial grappler or a sport grappler is an uneducated and irrelevant statement. The correct recipe for a champion in wrestling involves many hours of devoted physical conditioning, superior strength, speed, stamina, and the ability to out-work your opponent whether by sheer brawn and/or technique. The inclusion of a gi would be most unrealistic and pointless in a wrestler's training regimen. While many of the qualities mentioned would be beneficial to a jiu jitsu player, the training methodology and fostering of jiu jitsu skill requires a different approach.
For the sake of training a well-rounded Gracie Jiu Jitsu grappler that relies inherently on the principles of superior leverage and technique (which are qualities such that if they were taught to a person with no outstanding athletic or physical qualities, would be able to defend themselves and/or compete against stronger opponents), I believe the gi to be a mandatory element in a person's training.
Training with the gi offers an nigh infinite amount of variables with which the jiu jitsu player must concern himself in order to win. The gi is both dangerous and helpful to a player. Without it, you may not realize that your opponent has to ability (in both no-gi and gi) to choke you from a certain position that you consider to be safe. Training the body's reflexes, mind, and furthering the education of understanding which combination of variables, grips, and positions will result in the correct outcome allows for the player to become more technical.
One of Jigoro Kano's innovations in Judo was the refining of classical ju jutsu to Judo, where he excluded the practice of many dangerous and harmful techniques to katas. This allowed for Kano to further develop and improve on jujutsu training methods by focusing on randori with techniques that could be APPLIED safely to an opponent at full force, thus allowing for a safe way to improve physical and combat proficiency without maiming opponents.
I believe this principle can be directly applied to Gracie Jiu Jitsu grappling, where taking away the unreliable qualities of brawn and berserk allows for the fostering of safer and more effective qualities such as patience, superior timing, speed, technical knowledge, and true jiu jitsu prowess.