equals() is better choice to test equalitity of Strin object. Also, its faster that compareTo().
compareTo() compares two strings lexicographically.The comparison is based on the Unicode value of each character in the strings. The character sequence represented by this String object is compared lexicographically to the character sequence represented by the argument string.
equals() compares this string to the specified object. The result is true if and only if the argument is not null and is a String object that represents the same sequence of characters as this object.
You should ignore considerations about performance at this stage. There is probably no significant performance difference between the two; you can investigate the source code by unzipping the "src.zip" file in your Java installation folder. Have a look at this article by Brian Goetz and see what he says about performance. You should use methods for their designed purpose. If there is an equals() method use that for equality and if there is a compare/compareTo method use that for comparisons.
There are a few possible exceptions however. The BigDecimal equals method returns false if you present 1.0 with 1.00, so you would have to use the compareTo method and == 0.