The two words puberty and adolescence seem to be referring to one thing; what is the difference between them?
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."
- Refers to a biological process
- Can begin and end independent of adolescence
- May be described in terms of concrete physiological terms
- Refers to a period of time, usually teenage years though not always
- Will (practically) always overlap puberty while remaining distinct
- Described mostly in abstract psychological & sociological terms
Puberty does have some influence over mental maturity and development mainly through hormonal changes while adolescence generally speaks of personality and social maturity. It is common to notice that during adolescence there are certain physical changes such as a tendency to increase in height and changes in body shape but these are usually puberty's overlap or a continuation of the normal growth process.
Of these, I believe the main difference to be that puberty is a concrete biological processes where adolescence is an abstract social concept and the two do not always coincide in their entirety.
As reported by the Collins English Dictionary, puberty is "the period at the beginning of adolescence when the sex glands become functional and the secondary sexual characteristics emerge;" adolescence means "the period in human development that occurs between the beginning of puberty and adulthood."
Puberty is then used when referring to physical changes, and adolescence when referring to social and psychological changes.