The trust is a long existing (and still existing) legal concept by which property is held by one person(A) for the benefit of another (B) - i.e. A holds legal title, but B is to have the benefits from this. Trusts separate legal and beneficial rights on property, and are an entirely legitimate mechanism for holding and transferring property.

The corporations you mean became referred to as 'trusts' because they used the mechanism of a trust to hold together conglomerates of other corporations. At the time, a company itself could not own shares in another company, so the overall owners of the conglomerate would transfer all their different shares in different companies to a trust which they controlled (either personally or by proxy). Effectively it was to do what now can legally be done through a holding company.

The trusts were a mechanism of improving organisation and streamlining ownership, however many of the corporations using them were also associated with abusive business practices, collusion between large companies perverting the market, and monopolising - a prime example being Standard Oil. Monopolies were seen as undesirable (well they basically are, but thats an economics argument I don't want/not equipped to get into). The governments at the time wanted to do something because of the effects of these practices on the economy and consumer. 'Antitrust' law is essentially a US term for anti-competition law aimed at avoiding monopolies in the economy.

The basic difference would be legal structure - large corporations with many sub-corporations would be structured through a holding company as that is a much simpler way of doing things. Anti-trust and anti-competition laws also limit the sorts of abusive business practices that led to monopolisation and the emergence of anti-trust laws in the first place.

I cannot address Taft's role in trust busting, however, Theodore Roosevelt's reasons for going after these entities were based on his close relationship with both the environment and the American people. As a child, Roosevelt collected animal, insect, and plant specimens for classification in his many journals and diaries. During his college years, he left school to mourn the death of his first wife, moving out west to start a cattle ranch. He formed close relationships with migrants who settled on undeveloped land, and understood the daily struggle these people faced. The most pernicious obstacle for these settlers was the large corporations and railroads that simply took resources from the settlers' lands without compensation. When Roosevelt became an "accidental president" after the assassination of McKinley, he seized the opportunity to protect the environment and American citizens from unfair trust practices by setting aside huge swaths of land as national parks for future generations. Once called a "traitor to his class" by the wealthy trust owners, Roosevelt was considered a hero by his fellow citizens.

Sources:
Brands, H. W. American Colossus the Triumph of Capitalism, 1865-1900. New York, NY: Doubleday, 2010.
Brinkley, Douglas. The Wilderness Warrior Theodore Roosevelt and the Crusade for America. Pymble, NSW; New York, NY: HarperCollins e-books, 2009.
Donald, Aïda DiPace. Lion in the White House a Life of Theodore Roosevelt. New York: Basic Books, 2007.
Egan, Timothy. The Big Burn : Teddy Roosevelt and the Fire That Saved America. Boston: Houghton Mifflin Harcourt, 2009.
Morris, Edmund. The Rise of Theodore Roosevelt. New York: Modern Library, 2010.
Morris, Edmund. Colonel Roosevelt. New York: Random House Trade Paperbacks, 2011.
Thomas, Evan. The War Lovers Roosevelt, Lodge, Hearst, and the Rush to Empire, 1898. New York: Little, Brown and Co., 2010.
Zacks, Richard. Island of Vice Theodore Roosevelt’s Doomed Quest to Clean up Sin-Loving New York. New York: Doubleday, 2012.