The primary disagreement between the two parties was one of economic policy. Republicans traditionally have been advocates of supply side economics, sometimes called "trickle down" economics and more recently known as "Reaganomics" (although that term was not heard of at the time of the Great Depression.) Democrats tended (and still are) advocates of Keynesian Economics, a policy first promulgated by John Maynard Keynes of Oxford University.
Supply Side Economics is the philosophy that the economy can be stimulated by reducing taxes and providing incentives to businesses which will allow consumers to have more spending money. A classic (although much later) example was President George W. Bush's tax refund which gave consumers roughly $600.00 in tax refunds. The belief was this money would be spent which would stimulate the economy.
Keynesian Economics calls for massive government spending rather than tax cuts to stimulate the economy. The New Deal is a classic example. Under this philosophy, the government undertakes massive government projects (Such as the Tennessee Valley Authority and the Civilian Conservation Corps) which put people working directly for the government.
In either instance, the end result is always an increase in government debt, regardless of whether the intended stimulus of the economy is successful. This increase in spending is an easy target for political opposition; therefore regardless of the position pursued by either party, the other party will accuse the other of increasing government debt. This was the case during the New Deal, and has been ever since.