1. equals()

Returns true if they are same and not null.

2. equalsIgnoreCase()

Returns true if they are same ignoring Case(Upper or Lower Case) and not null.

Sample Code:

String str1 = 'abc';
String str2 = 'ABC';
system.debug( str1.equals( str2 ) );
system.debug( str1.equalsIgnoreCase( str2 ) );

Output: