It is up to the head daimyō to decide. Usually he himself at least commanded the main force, while he could dispatch generals with separate forces. But there are cases where the daimyō just sat at home and delegated out the military leadership. But doing so always runs the risk of your commander being less than loyal.

The norm in the Sengoku was that important tasks, like military command, were given to 譜代 fudai, important samurai families that had served the daimyō loyally for multiple generations, or trusted relatives. The biggest and most well known exception to this is Oda Nobunaga, who seemed to have looked at talent predominantly and paid little attention to traditional ties. That's not to say other daimyō did not promote based on merits, but often some other kind of ties or established power base is considered.

Both vassals and retainers are imperfect terms borrowed from Medieval European historiography to describe Japan in terms more familiar to a western audience. As such, they are imperfect. So I would say we can't really say how the terms are different. The Japanese term is 家臣 kashin, which means anyone in the clan's service. It is further divided into, 旗本 hatamoto - families under the direct employment of the clan, 譜代 fudai - families with a long history and close ties with the clan, and 外様 tozama - families with not-so-close ties with the clan.