1) Debt matters because it smoothes (or unsmoothes) taxes over time, which matters because the deadweight loss from taxation is (roughly) proportional not to the tax rate, but to the square of the tax rate.
2) Because family sizes are not homogeneous, government debt (which transfers the tax burden to future generations) can redistribute the tax burden across families. (Think, for example, about a taxpayer with no kids, or a family that has just immigrated.)
3) Government debt makes it easier to borrow --- it's better to pay 3% to government bondholders than 18% to your credit card company. This is possible because the penalty for not paying your taxes is considerably more severe than the penalty for not paying your credit card bill. (In effect, government debt resurrects the institution of the debtor's prison.)
4) There are of course issues with misperceptions, where debt can make people feel either richer or poorer than they really are, depending on whether they underestimate or overestimate their future tax burdens, and so can affect consumption paths in either direction.