Learn English – Difference between “theorem” and “theory”

In short: puberty is a period of physical transition, adolescence is about a psychological and social transition.

Puberty is the process of physical changes by which adolescents reach sexual maturity, i.e. become capable of reproduction. Puberty refers to the bodily changes, while adolescence is the period of psychological and social transition between childhood and adulthood. (sources: Wikipedia and New Oxford American Dictionary).

In nonspecialized discussion and writing, the meanings tend to get blurred, and puberty can be used to refer to the period during which the changes happen, getting a meaning closer to adolescence. Adolescence still encompasses a typically larger period of time, however: puberty is over when the young individual's body has fully transformed, while it takes some more time for him or her to be recognized as an adult. Criteria for the latter are somewhat arbitrary, the typical example being “having reached the age of majority or being emancipated” (the age of majority depending on the country of residence). Some people even talk of 20-something adolescents to refer to people who have not yet reached an autonomous state of living, while being of full legal age.

As a side note, adolescence has a related noun (adolescent) and adjective (also adolescent), while puberty has a related noun (pubescent) and three adjectives (pubescent, pubertal, and sometimes puberal). It also has a synonym, "pubescence."

Yes. Continuous means from some start to some end without break:

The water flowed continuously over the dam.

whereas continual means occurring repeatedly at intervals over a time span:

I continually lose at poker.

I should also mention that continual is often substituted for continuous, and would be correct in most contexts, however the converse is not generally true. That is to say,

The water flowed continually over the dam.

is okay, but you would (hopefully) not mean

I continuously lose at poker.

since that would imply that all you do all day long is play poker and lose.