In due time MoneyGram separated from the Western Union and became an independent brand. What are the main benefits of money transfers by MoneyGram? Cost. Today, it is most advantageous to the customer service of all the similar services. MoneyGram completed the rollout of simplified pricing for money transfer services in the U.S., an initiative that began last year. Today, MoneyGram - is the second world leader by the growth of money transfers between individuals and the dynamics of international network services.