1 External debt is owed to foreign countries
External debt is an amount of money owed to foreign countries or governments. If a country borrows money from another country, this will be considered external debt.
2 Public debt is owed to citizens
Public debt is an amount of money usually owed to citizens and institutions who lent the government money by buying governmental bonds, for example.
3 External debt is usually in foreign currency
A country's external debt will usually be in foreign currency, whereas the public debt will be in the country's own currency. Countries can thus print more money to repay public debt.
4 External debt is more dangerous
An internal debt can be dealt with by a government by printing money because it will have to repay it in local currency. External debt usually represents a bigger problem, as it usually has to be repaid in a foreign currency. (See What happens when a country has too much debt?)
5 Public debt can be owed to foreigners (special case)
Some governments allow foreigners and foreign institutions to buy their government bonds in local currency. In such a case, some of the public debt will be owed to those foreigners.
6 High debt isn't always bad
The debt alone can't tell whether a country is doing well or bad. If the economy is going well, the debt to GDP ratio might not be a problem even if it is relatively high. On the other hand, bad economy in addition to a high debt constitute a serious problem. (See Why do countries devalue their currency?)