== tests for reference equality.

.equals() tests for value equality.

== is for testing whether two strings are the same object.

Comparison symbols is only can use by local variable only.
Public Variable should use "equals" to compare.

Java

@Kroll.method
    public void ComparisonTest(String value)
    {
        String operation = value;
        String operation_local = "+";


        //get result true
        if (operation.equals("+"))
        {
            System.out.println("equals = true");
        }else{
            System.out.println("equals = false");
        }

        //get result false
        if (operation == "+")
        {
            System.out.println("comparison = true");
        }else{
            System.out.println("comparison = false");
        }

        //get result true
        if (operation_local.equals("+"))
        {
            System.out.println("local_equals = true");
        }else{
            System.out.println("local_equals = false");
        }

        //get result true
        if (operation_local == "+")
        {
            System.out.println("local_comparison = true");
        }else{
            System.out.println("local_comparison = false");
        }

    }

JavaScript

Ti.API.info("Input Test = "+calculatorModule.ComparisonTest("+"));  

Refer:
http://stackoverflow.com/questions/513832/how-do-i-compare-strings-in-java