A lot vem no fim da frase "they like to ski a lot" "thanks a lot", significa "muito" e passa ideia de intensidade
a lot of vem no meio da frase "they have a lot of books" "they have a lot of money" , significa "muito" ou "muitos" e passa a ideia de quantidade
a lot of pode tbm ser substituido por lots of "they have lots of books" "they have lots of money"