Just like the most, if not all, other pairs of originally Japanese words and their Sino loanword counterparts, the former ([優]{やさ}しい in this case) is more intuitive in meaning and/or nuance than the latter ([親切]{しんせつ}). Japanese-speakers learn the word 「優しい」 a few years before they get to learn 「親切」.
The biggest difference between the two words, IMHO, is that while 「優しい」 expresses an innate human quality,「親切」 is mostly used to refer to an acquired characteristic.
If I tried to give you the English counterparts for the two words, it would probably confuse you just like a bilingual dictionary (or a textbook in your case) would confuse you, but I shall try.
If you are an innately "gentle" person, you are then at least a 優しい person. You have it; It was not really taught.
Does that automatically make you a 親切な person? No, it does not. To be called a 親切な person, you need to "learn" to show it in your deeds. This is why I called 親切 an "acquired" quality. "Considerate in a friendly way", perhaps?
Example:
You asked someone for directions on the street and he drew a map to show you how to get where you needed to. You call that 親切 because he took the time to help you on a hot and humid summer day.
Whether or not he is 優しい inside, you will not know until you have spent more time with him. In fact, whether he is 優しい or not does not matter because he is a stranger and the chances are that you will never meet him again even if you may remember him for a long time as an extremely 親切な人.
There is even a possibility that he might be the most 優しくない人 in the world in his private life. But what do you care? He helped you immensely out on the street on that day!
