Moths fly both during the day and the night. There are many different species of moth and each has different habits and food preferences. The question is too vague to be meaningful. Which moths are you asking about?
Not all species of moths are nocturnal. Only the nocturnal ones rest during the day and come out at night. Moths and animals that are active during the day and rest at night are referred to as diurnal.
One way butterflies are different from moths: is that butterflies are active during the day while moths are active at night. Another one is that butterflies differ in color while moths are typically one color.