Why was the Bull Moose Party formed?

The basic reason why the Bull Moose Party was formed is that Theodore Roosevelt wanted to get elected president in 1912.  He was upset with the way that the Republican Party had gone after he left office in 1908 and he wanted to run against President Taft.

Roosevelt was the first of the progressive presidents.  When he was planning to leave office in 1908, he picked William Taft as his successor.  He believed that Taft would continue his progressive policies.  However, Taft's presidency did not go the way TR wanted it to.  Although Taft did many progressive things, he did not satisfy TR.  Among other things, he fired one of TR's proteges, Gifford Pinchot, a staunch conservationist.  Most historians argue that Taft really was no less of a progressive and they say that TR's main motive for opposing Taft was ego.

So, the Bull Moose Party formed because TR disliked what Taft had done and he wanted to be president again.

How was the Bull Moose party different from its contemporary Republican Party?

The Bull Moose, or "Progressive" Party is sometimes held up as an example of a "bolter" party, i.e., one that leaves a major party to field candidates on its own. The Bull Moose Party was formed when Theodore Roosevelt left the Republicans in advance of the 1912 presidential election. Essentially, he claimed that William Howard Taft, the Republican nominee, had backtracked on the Progressive agenda that Roosevelt had brought to the White House. The Bull Moose Party thus differed from the Republicans in that they embraced far more reforms than the more conservative (but not entirely reform-averse) Republicans under Taft. Roosevelt's platform, often called the "New Nationalism," was for its time one of the more radical political positions ever taken by a Presidential candidate. It called for strict regulation of the trusts, woman suffrage, outlawing child labor, an eight-hour workday, currency reform, an income tax, and for transparency in investments, among many other initiatives. Most of these reforms were anathema to the Republicans, who embraced only limited regulations on the economy.