I was reading a book from Dale Carneige named "The Art Of Public Speaking"(Originally published: 1915). I encountered that word "annals" is that exactly same with "annuals" or are there tiny differences?
annals means reports or records, as of a society or club, or of a broad topic: Annals of Medicine (a medical journal) or annals of history (a literary term meaning "recorded history"). The etymology shows that it comes from the Latin annus, "year," though its usage today is not restricted to things that are published only once a year. The word is always plural.
annual, as a noun, means a yearly publication (or yearly version of a publication) and also comes from the Latin annus. In that sense it seems rather similar to annals, at least in the context of the reports of a society/club/etc, but it does apply only to things published once annually, as "annals" does not. Here is a page showing a publisher's annuals; you can see they are meant for the mass consumer market, unlike the reports or minutes of a society that are published for posterity's sake. This usage appears to be chiefly British; as an American English speaker I have never heard of this usage before now, and when you say "annual" I would first think of the botanical meaning.