String.format is just a shortcut to Formatter, this is a "printf-style" formatter. On the other side, MessageFormat uses a different formatting convention, as described in the linked documentation.
Use the first "for layout justification and alignment, common formats for numeric, string, and date/time data, and locale-specific output" and the second "to produce concatenated messages in language-neutral way".
Put simply, the main difference is in format string:
MessageFormat.format() format string accepts argument positions (eg. {0}, {1}). Example:
"This is year {0}!"
The developer doesn't have to worry about argument types, because they are, most often, recognized and formated according to current Locale.
String.format() format string accepts argument type specifiers (eg. %d for numbers, %s for strings). Example:
"This is year %d!"
String.format() generally gives you much more control over how the argument is displayed thanks to many options you can specify with the type specifier. For instance, format string "%-6.2f" specifies to display a left-aligned floating point number with min. width 6 chars and precision of 2 decimal places.
Just have a look at javadoc of both methods to find out more details.