Furnish, equip, outfit, appoint, accouter, arm are comparable when they mean to supply a person or something used by him with the adjuncts necessary or appropriate to his daily living or his occupation.
Furnish stresses the provision of all essentials; thus, a house is furnished when it is supplied with all the necessary conveniences that make it ready for use as a home.
Equip stresses the provision of things or occasionally of a single thing making for efficiency in action or in use; thus, a poorly furnished kitchen may be short in tables or chairs, but a poorly equipped kitchen is not adequately provided with the utensils or appliances needed for cooking and other work carried on there; one equips, rather than furnishes, an automobile with a four-wheel drive.
Outfit stresses provision for a journey, an expedition, or a special kind of activity (as work); it is used chiefly with reference to necessary clothes, tools, utensils, and accessories and so is narrower in its range of applications than furnish and broader than equip.
Appoint, which is somewhat bookish in this sense, suggests complete and often elegant furnishings or equipment.
Accouter stresses provision of dress, array, or personal equipment, usually for a particular activity (as military service).
Arm stresses provision for effective action or operation; it is used chiefly with reference to equipment necessary for offense or defense, but it may imply no more than provision of a means of preparation for added strength or security. In their extended senses the words in this group retain their respective implications but refer to mental, moral, or physical qualifications rather than to things.