There is always some difference. "Проснуться" is more common in ordinary life. However, each time you encounter words with similar meaning, the context may slightly differ, or one can be older, or maybe one is colloquial and one bookish. Finally, the two may be really close, but in some idioms only one of them can be used.
It would be easier if you used a beginner's textbook, this way you'll be guaranteed to learn the more common words first. I mean, they may not be more often IN GENERAL, but more often used within a certain context than other equivalents.
You may also check frequency list to roughly get the idea how much do you need a certain word
落碼 ?髓銜逶 驪鈞瑁
For example, сильный ("strong") is in the first 500, while "могущественный" (potent, powerful) lags around 10 000.
Проснуться is 1914th lemma, and is used twenty times as often as "пробудиться" (place 1665.
** It is a dictionary based on Russian National Corpus, i.e. the word frequency was analyzed based on a huge amount of texts in ALL styles: literature, speech, newspapers, e-mails, interviews, movies etc. Don't expect it to adequately represent the frequency of such words as #&@, #*$&&, *!^#% or "convolution", the first three being thousands more popular in real-word debates than in newspapers or legal documents (same as in English, basically), and the latter never really used outside of mathematical or programming literature. I mean, the frequency is going to be right - on average, so sometimes among rare words there are diffirent kinds of rareness. For example, you'll find that "convolution" (свертка) is, in fact, slightly more popular thaт "to be jumping onto something" (запрыгивать). However, one of two you are not likely to ever encounter.