What is the meaning and application of the apostrophes in derivation paths of BIP32 / BIP44? Can we please clarify if there is any difference (because apparently some technical documentation explicitly requests to include apostrophes) between a derivation from extended private key (xprv) or public key (xpub) of a child subkey, where m = root (xprv or xpub).

m/44/0/0/0 and  m/44'/0'/0'/0

What are the meaning of the apostrophes and how / when should they be applied? Do they change anything in a derivation path -- will the derived child subkey be different with or without them?
The apostrophes show whether or not that particular derivation is hardened. 

The difference been hardened and non-hardened keys is described in BIP32.

See the wiki for some further detail. https://bitcoin.org/en/glossary/hardened-extended-key