Conceding in the legal world, as in the real world, goes against human nature. It usually consists of acknowledging a fact or point of law that goes against your client's case. However, when properly used, it conveys the savvy-ness of an attorney. It indicates that an attorney has taken their blinders off and has considered the other side's point of view. It also indicates a modicum of humility.
No one likes to admit their weaknesses. Lawyers are no exception. The adage of "lead with your best argument" is still certainly true. However, there is no such thing as a perfect case and/or client. Lawyers better serve their clients by addressing weaknesses after presenting their best argument. To do this shows that you fully understand the issues. It also potentially takes away the sting of the opposing attorney stating your weaknesses when you blindly ignores them.
I recently dealt with an attorney who distorted our client's deposition in a motion to dismiss that he filed with the court. Not only did he misrepresent our client's testimony, he made false statements in the actual motion. Countering his motion was simple: state what our client actually said. This lawyer adopted such a one-sided view that he allowed it to cloud his judgment. After I brought up the many weaknesses of his argument in a responsive motion, he dropped his motion and decided to settle the case. If he had simply conceded certain things that were true, he may have been able to sting our client more effectively.
Refusing to admit weaknesses makes you appear conceited. Concede and leave your reputation intact.
