I completely understand why a regular homotopy is required to pass through only immersions of the first manifold into the second, but I am not confident I understand why the second condition--it must extend continuously to a homotopy of the tangent bundles--is imposed. Are there any good examples to help me understand the presence of this second condition? What undesirable results would follow were this condition dropped?
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Regular homotopy needs a condition that ensures that the derivatives of the curve are continuous. Some authors formulate this by saying the homotopy is smooth but there seem to be several other ways to do it, one of which is the formulation of Smale.
Without this condition the Whitney degree of a circle immersed in the plane is not conserved. See the discussion in: A contact geometric proof of the Whitney-Graustein theorem http://arxiv.org/pdf/0801.0046 by H Geiges.