Indolence means “laziness” or “relaxation” while insolence means “being rude to a superior or person of higher class.”
Here's a quick look at indolence used in a sentence:
His indolence prevented him from financial success.
It also has a common adjective form, which is indolent:
We enjoyed a wonderfully indolent weekend on the beach.
In other words, indolence can have the negative meaning of laziness, or the positive meaning of relaxation with the connotations of luxury and leisure.
Insolence is always negative—at least to the target of the insolence. Here are a couple of sentence examples to illustrate:
She called her dominating boss a rude name and was fired for her insolence.
The insolence of comedians to authority is often admired.
Although it is negative to call someone insolent, or to refer to a person’s insolence, in American culture, the person who calls someone insolent will be seen more negatively than the insolent person—because Americans tend to believe that all people should be seen as equals. People who use the word insolence seriously may be seen as “stuck-up” or “putting on airs” (acting as if they are more important than they really are).