Exactly what FileUtils.contentEquals method of Apache commons IO does and api is here.

Try something like:

File file1 = new File("file1.txt");
File file2 = new File("file2.txt");
boolean isTwoEqual = FileUtils.contentEquals(file1, file2);

It does the following checks before actually doing the comparison:

• existence of both the files
• Both file's that are passed are to be of file type and not directory.
• length in bytes should be the same.
• Both are different files and not one and the same.
• Then compare the contents.

Your program's time complexity is \$O(n*m)\$ and space complexity is \$O(n + m)\$ where 'n' is no. of lines in the first file and 'm' is no. of lines in second file.

Here is an optimised version of the above program, with time complexity \$O(n + m)\$ and space complexity \$O( min(m,n) )\$. I have not tested this program, but it should be able to present output on the screen within few seconds :)

import java.io.*;
import java.util.*;

class Main{
   public static void main(String args[]){
      try ( BufferedReader reader1 = new BufferedReader(new FileReader("file1.txt"));
            BufferedReader reader2 = new BufferedReader(new FileReader("file2.txt")) ){

            //assuming file1.txt is smaller than file2.txt in terms of no. of lines
            HashSet<String> file1 = new HashSet<String>();

            String s = null;
            while( ( s = reader1.readLine()) != null){
               file1.add(s);
            }

            while( (s = reader2.readLine()) != null ){
               if(file1.contains(s))
                  System.out.println(s);
            }
      }
      catch(IOException e){
         System.out.println(e);
      }

   }
}

Note: Only one file is in memory at a time and HashSet<> instead of nested loops for comparison.

I converted your code into a main program. There is no infinite loop in this code.

I am assuming you are comparing 2 text files of a small-ish size.

import java.io.*;

public class Diff {
    public static void main(String[] args) throws FileNotFoundException, IOException {

        File f1 = new File(args[0]);// OUTFILE
        File f2 = new File(args[1]);// INPUT

        FileReader fR1 = new FileReader(f1);
        FileReader fR2 = new FileReader(f2);

        BufferedReader reader1 = new BufferedReader(fR1);
        BufferedReader reader2 = new BufferedReader(fR2);

        String line1 = null;
        String line2 = null;
        int flag = 1;
        while ((flag == 1) && ((line1 = reader1.readLine()) != null)
                && ((line2 = reader2.readLine()) != null)) {
            if (!line1.equalsIgnoreCase(line2))
                flag = 0;
        }
        reader1.close();
        reader2.close();
        System.out.println("Flag " + flag);
    }
}

I ran it on 2 small different text files. This is the output.

javac Diff.java && java Diff a.txt b.txt
Flag 0

If you think you have an infinite loop, the issue might be elsewhere.

You could iterate over these two files line by line using a BufferedReader

the readLine() method returns a string representation of each line which can then be checked for equality.

The index of any unequal lines can be stored and returned in any way you want.

The below code will serve your purpose irrespective of the content of the file.

import java.io.BufferedReader;
import java.io.FileNotFoundException;
import java.io.FileReader;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.util.ArrayList;
import java.util.List;

    public class Test {
        public Test(){
            System.out.println("Test.Test()");
        }

        public static void main(String[] args) throws Exception {
            BufferedReader br1 = null;
            BufferedReader br2 = null;
            String sCurrentLine;
            List<String> list1 = new ArrayList<String>();
            List<String> list2 = new ArrayList<String>();
            br1 = new BufferedReader(new FileReader("test.txt"));
            br2 = new BufferedReader(new FileReader("test2.txt"));
            while ((sCurrentLine = br1.readLine()) != null) {
                list1.add(sCurrentLine);
            }
            while ((sCurrentLine = br2.readLine()) != null) {
                list2.add(sCurrentLine);
            }
            List<String> tmpList = new ArrayList<String>(list1);
            tmpList.removeAll(list2);
            System.out.println("content from test.txt which is not there in test2.txt");
            for(int i=0;i<tmpList.size();i++){
                System.out.println(tmpList.get(i)); //content from test.txt which is not there in test2.txt
            }

            System.out.println("content from test2.txt which is not there in test.txt");

            tmpList = list2;
            tmpList.removeAll(list1);
            for(int i=0;i<tmpList.size();i++){
                System.out.println(tmpList.get(i)); //content from test2.txt which is not there in test.txt
            }
        }
    }

Your code should be

Scanner input1 = new Scanner(new File(firstName));//read first file
Scanner input2 = new Scanner(new File(secondName));//read second file

while(input1.hasNextLine() && input2.hasNextLine()){
    first = input1.nextLine();   
    second = input2.nextLine(); 

    if(!first.equals(second)){
        System.out.println("Differences found: "+"\n"+first+'\n'+second);
    }
}

// optionally handle any remaining lines if the line count differs

Previously you only compared one time, the very last line. But you need to compare after each line you read.