EXL

Попробуйте привести строки к одной общей кодировке, а затем сравнить. Для этого можно воспользоваться библиотекой libiconv:

main.cpp:

#include <iostream>
#include <fstream>
#include <cstdlib>

using namespace std;

#include <iconv.h>

string iconv_recode(const string from, const string to, string text)
{
    iconv_t cnv = iconv_open(to.c_str(), from.c_str());

    if (cnv == (iconv_t) - 1) {
        iconv_close(cnv);
        return "";
    }

    char *outbuf;
    if ((outbuf = (char *) malloc(text.length()*2 + 1)) == NULL) {
        iconv_close(cnv);
        return "";
    }

    char *ip = (char *) text.c_str(), *op = outbuf;
    size_t icount = text.length(), ocount = text.length()*2;

    if (iconv(cnv, &ip, &icount, &op, &ocount) != (size_t) - 1) {
        outbuf[text.length()*2 - ocount] = '\0';
        text = outbuf;
    } else {
        text = "";
    }

    free(outbuf);
    iconv_close(cnv);

    return text;
}

void compare_strings(const string &aString1, const string &aString2) {

    cout << "String 1: " << aString1 << endl
         << "String 2: " << aString2 << endl;

    if (aString1 == aString2) {
        cout << "Identical strings!" << endl
             << "-----" << endl;
    } else {
        cout << "Different strings!" << endl
             << "-----" << endl;
    }
}

int main()
{
    ifstream file_1("word_1.txt");  // The "Proverka" Word in UTF-8
    ifstream file_2("word_2.txt");  // The "Proverka" Word in CP1251
    string word_1, word_2;

    file_1 >> word_1;
    file_2 >> word_2;

    compare_strings(word_1, word_2);

    word_2 = iconv_recode("CP1251", "UTF-8", word_2);

    compare_strings(word_1, word_2);

    return 0;
}

exl@exl-Lenovo-G560e:~/SandBox/text_enc > enca -L russian  word_1.txt 
Universal transformation format 8 bits; UTF-8
  Doubly-encoded to UTF-8 from ISO-8859-5
exl@exl-Lenovo-G560e:~/SandBox/text_enc > enca -L russian  word_2.txt 
MS-Windows code page 1251
  LF line terminators
exl@exl-Lenovo-G560e:~/SandBox/text_enc > cat word_1.txt 
Проверка 
exl@exl-Lenovo-G560e:~/SandBox/text_enc > cat word_2.txt 
��������
exl@exl-Lenovo-G560e:~/SandBox/text_enc > ./text_coding 
String 1: Проверка
String 2: ��������
Different strings!
-----
String 1: Проверка
String 2: Проверка
Identical strings!
-----