The quick answer
The term Insolvency is the term to cover all types of debt problems while Bankruptcy is the term for an individual (whether in business or not) who has been declared bankrupt. Therefore bankruptcy is a type of insolvency. Other types of insolvency for individuals include IVA’s, debt relief orders and debt management plans.
In more detail
Bankruptcy is the term for when an individual is declared bankrupt by the Court because they are insolvent. People go bankrupt for two reasons; one is if their assets are less than their liabilities (so they owe more than they own) and the second is where they are unable to pay their debts when they fall due.
A Court will satisfy itself that the person being made bankrupt fits one of these two conditions before it makes a bankruptcy order.
Insolvency is the generic term for all types of insolvency including: liquidation, administration, bankruptcy, voluntary arrangements and debt relief orders.