On Wednesday, in a piece trying to set out the reasonable case for austerity, I made one argument against government deficits—they crowd out the private investment that really raises our living standards over time. I think this is true outside of exceptional circumstances when the economy is in a slump and the central bank is unwilling (or perhaps unable) to rectify the situation. But there is an important reason why raising taxes isn't a simple solution to the problem. Except for lump-sum taxies levied on natural ability (impossible) or existence (unpopular and possibly unfair) the government doesn't have access to any tools for raising money that don't distort the economy. Higher taxes lead to lower economic activity; lower taxes lead to higher economic activity.
If you or I or a firm incurs a debt, we (usually) have to work hard to create wealth to pay it back. That is, increasing private debt does not reduce economic activity. If it's used to fund consumption it's likely to increase it a smidgeon—if it's used to fund investment it's likely to increase it somewhat.
This is different when the government incurs debt. It sells a bond, which on the margin must be slightly more attractive than all other existing investment products, thus creating a bit of extra value for investors/savers. It eventually must pay these back through raising taxes or monetisation. Monetisation is bad, creates inflation or even hyperinflation, and thankfully in developed countries governments do not do this.
A country can 'grow its way out of debt' but only in the sense that the taxes required to pay the debt off are smaller relative to the size of the economy. £10bn less debt is still £10bn less (plus interest) in taxes. Eventually all debts must be paid off with extra taxes, whether you do them now or later.
These taxes reduce economic activity through their deadweight losses (presuming we're already levying efficiency-increasing optimal Pigovian taxes to fund normal expenditure). This means that extra government debt will lead to less activity than there otherwise would have been some time down the line, over and above its crowding-out effect on investment.
There's an interesting side-point here comparing tax-funded or deficit-funded spending overall. Should we borrow now and tax later or run balanced budgets all the time? This is another question—but my point is that either way we fund spending it's going to be costly.