involved (in sth) usually means taking part in something; being part of something.
He was not involved in the conspiracy.
I want to meet the person involved in this project.
A company's HR department may also be involved in making people redundant. It implies department is part of or takes part in the process of making people redundant.
involved(with somebody/something) means giving a lot of time or attention to somebody/something.
She was deeply involved with the local hospital.
He has been involved with human rights for many years.
The HR department will also be involved with training and professional development of the company's staff. It implies the department will give a lot of attention to the development of the company's staff.