This is a typical imperfective/perfective pair of verbs, they are different in their aspect only. And they, actually, have the same root, -пад-, only in падать it is followed by a thematic vowel -a-, and упасть does not have the vowel, the root is immediately followed by the infinitive suffix -ть, so the two similar consonants get dissimilated:
*у-пад-ть -> упасть
Still, in its Future forms the final д of the root reappears:
я упаду
ты упадёшь
он упадёт, etc.
And in the Past forms this д disappears again, before the Past Tense suffix -л:
*у-пад-л -> упал
As for the meaning of the two verbs, падать is the name of the process of falling, "to fall/to be falling", and also it means "to fall repeatedly, all the time":
Падают листья. - The leaves fall/are falling.
Я всё время падаю, когда катаюсь на коньках. - I'm always falling while skating.
Упасть also means "to fall", the difference is it implies falling as having either the beginning or the end of it. Since things usually fall fast, this very verb is rarely used in the meaning "to begin falling", it mostly means "to fall down (and reach the bottom or the lowest level)":
Он упал и сломал ногу. - He fell down and broke his leg.
Цена на нефть упала. - The price of oil has dropped.
Note that упасть, like all the perfective Russian verbs, cannot be used in the Present Tense, its forms that look like Present actually mean the Future. Падать can be used in all the three Russian tenses.
