my short answer is no, there is a difference between these.
if you take the example of european countries, you could see that those which are located on the western part, such as the uk, france or germany - these are called developed countries with firm economies, a great gdp, strong currency, etc.
however, if you go east, you can find countries such as hungary, slovakia and romania, which are "just" deeloping countries - they need some financial aid from other countries, their economies are not as stable as the above mentioned countries', they have a higher unemployment rate, weaker currencies, etc.
personally, i have only heard the terms developing / developed so i wouldn't use advanced / advancing.