When "incompetent" meets "inept": a dysfunctional duo

What is similar?
Both "incompetent" and "inept" describe a lack of skill, ability, or proficiency in performing a task or function.
What is different?
While both words indicate a lack of skill, "incompetent" often refers to lacking the necessary qualifications or competence to perform a task successfully, especially in a professional or formal context. "Inept" often implies clumsiness or awkwardness and can suggest a lack of judgment or tact, especially in social situations.
Which one is more common?

Examples of usage
Incompetent- The manager was fired because he was incompetent at running the department.
- Due to his incompetent handling of the project, the company lost a major client.
- She felt incompetent after failing the certification exam for the third time.
- His inept attempts at cooking always ended in disaster.
- The comedian's inept delivery of the joke left the audience confused.
- She was socially inept and struggled to make friends at the new school.