For paper wallets generated at bitaddress.org, are the public keys and BTC addresses the same thing?
No, they're not the same thing. Paper wallets are not generated differently as/in to comparison to normal wallets.
a bitcoin adress is part of a public key.
The address is at its most basic just a hash of the public key. The hash functions involved (RIPEMD-160 and SHA256) are cryptographic hash functions. They are often also referred to as one-way functions, which is exactly the reason why you cannot derive the public key from the address.
Oneway functions provide exactly that a = f(pk) so that given a you cannot derive pk.
So, an adress is basically the hash of a public key. x/x
Some good more indepth explanations of the differences here -> https://www.reddit.com/r/Bitcoin/comments/3filud/whats_the_difference_between_public_key_and/ctp3fc7/?st=jald5nmq&sh=620d5934
And, how the adress is derived from the public key is explained here ->