Adding to the states formed by splitting from other states are Alabama and Mississippi which were once part of Georgia. And they are no worse for the wear for having split off, so what's the problem with Scotland doing the same?
Answer: Asymmetric shocks. When the EU was formed, everyone (except economists) were happy. In 2009 many southern European countries faced debt and balance of trade crises. The normal response is to devalue your currency, juice exports, get the economy going, pay down debts, etc. But Spain, Greece et al. couldn't do that because they were locked into a common currency.
If the economic futures of Scotland and England fail to track each other (something that is likely), Scotland will be vulnerable to the same consequences of an asymmetric shock. Scotland will be unable to adjust its currency to deal with the problem. Like Greece and Spain they will have voluntarily given up an important economic tool. There is nothing very controversial about any of that. The importance of being able to adjust your currency is received macroeconomics wisdom.
Note that in the US all states use a common currency, but the heavy fiscal lifting is done by a common government - the feds. Thus asymmetric shocks that hit, say, Georgia, but that don't hit other states, are remedied any number of ways by the federal government.
Scotland would not have that escape hatch. It would be the worst of both worlds for Scotland. First, English taxpayers are not going to pay to bail out an independent state and second, Scotland will have given up an effective (arguably the most effective) economic tool to deal with its problems. Note what Germany is saying about Greece these days. It is a dress rehearsal for what England will be saying about Scotland if it ever gets in trouble.
Bob