The five sons of Mayer Amschel Rothschild were placed on the first rung of the nobility by the Austrian Emperor in 1817. They were granted the right to armorial bearings, and to use the suffix 'von' in their names. Those members of the family living in Frankfurt and Vienna were thus known as von Rothschild, whereas the Paris, and later the Naples, branches of the family adapted this to the French style 'de' - James de Rothschild, and so on. Their titles were enhanced to 'Baron' in 1822.
In England, Nathan Mayer Rothschild eschewed 'foreign' titles, which anyway would have counted for little. He is reputed to have declared that "plain Mr Rothschild" was good enough for him. His sons thought differently, and successfully applied to use the Austrian title, hence Baron Lionel de Rothschild - using the French style rather than the German one.
Lionel's son, Nathaniel or Natty, became the first Jewish peer in 1885. Somewhat unusually, he chose his own name as his title, and became Lord Rothschild. Thereafter in England, descendants of Lord Rothschild have not used the suffix - Miriam Rothschild, for example - whereas descendants of the first Lord Rothschild's brothers have continued to use 'de' - Edmund de Rothschild, for example.