Both have a similar meaning but there is a slight difference. "However" is a more flexible word that can be used to contrast information from positive to negative, or negative to positive.
Henry was a bad soccer player. However, he enjoyed playing with his friends.
Lionel Messi is the best soccer player in the world. However, he has never won a world cup
"Nevertheless" really only works to contrast a positive from a negative. Therefore, I could use it to replace "however" in the first example, but it would feel unnatural in the second. It is closer in meaning to "despite."
Henry was a bad soccer player. Nevertheless, he enjoyed playing with his friends.
The "higher level" would probably be "nevertheless."
The most important thing in writing is that you use a variety of transition words and phrases and mix up your sentence structure.
Let me know if you have any more writing questions!