[amsfonts] is a package provided by the AMS. It includes fonts with additional symbols and alphabets (blackboard bold, fraktur and several Euler scripts).
amsfonts is a package provided by AMS that includes additional symbols and alphabet styles. It is included on CTAN and in TeX Live.
The symbols are designed to be compatible with those in the Computer Modern fonts.
To use non-alphabetic symbols, it is necessary to specify \usepackage{amssymb}.
amssymb.sty is separate owing to the historical need to save space; it is part of the amsfonts bundle.
Additional alphabets are provided for use in math: blackboard bold (uppercase only, except for \Bbbk, a lowercase "k"), fraktur, Euler script (uppercase only) and Euler cursive (full alphabet with matching greek).
For documentation, see texdoc amsfonts on a system based on TeX Live.
