"Перекусить" = "to eat (usually) small amount of food between or instead of breakfast/lunch/dinner, to supress hunger". Something like "немного, но достаточно, поел". "- Будешь обедать? - Нет, спасибо. Я перекусил по дороге сюда."
note "переесть" = "to overeat".
"Закусить" = "to eat (usually) small amount of food during alcohol consumtion, to clear taste, to soften drunkness and so on". It is associated with alcohol. However, term "закуска" is a little more neutral and similar to "snack", but anyway food producers prefer to adopt term "snack" - "снек" instead of "закуска".
note: "заесть" = "to eat some kind of food immediately after another (often to soften some effects of it, for example bitter or strong taste)". It is not associated with alcohol.