Ibn `Abbas said that,
وَالَّذِينَ يُؤْمِنُونَ بِمَآ أُنزِلَ إِلَيْكَ وَمَآ أُنزِلَ مِن قَبْلِكَ
(And who have faith in what is revealed to you and in what was revealed before you.) means, "They believe in what Allah sent you with, and in what the previous Messengers were sent with, they do not distinguish between (believing) them, nor do they reject what they brought from their Lord."
وَبِالأْخِرَةِ هُمْ يُوقِنُونَ
(And in the Hereafter they are certain) that is the resurrection, the standing (on the Day of Resurrection), Paradise, the Fire, the reckoning and the the Scale that weighs the deeds (the Mizan). The Hereafter is so named because it comes after this earthly life.
The people described here (2:4) are those whom Allah described in the preceding Ayah,
الَّذِينَ يُؤْمِنُونَ بِالْغَيْبِ وَيُقِيمُونَ الصَّلٰوةَ وَمِمَّا رَزَقْنَٰـهُمْ يُنفِقُونَ
(Those who have faith in the Ghayb and perform Salah, and spend out of what we have provided for them.)
Mujahid once stated, "Four Ayat at the beginning of Surat Al-Baqarah describe the believers, two describe the disbelievers, and thirteen describe the hypocrites." The four Ayat mentioned in this statement are general and include every believer, whether an Arab, non-Arab, or a person of a previous Scripture, whether they are Jinns or humans. All of these attributes complement each other and require the existence of the other attributes. For instance, it is not possible that one believes in the Unseen, performs the prayer and gives Zakah without believing in what the Messenger of Allah ﷺ and the previous Messengers were sent with. The same with certainty in the Hereafter, this is not correct without that, for Allah has commanded the believers,
يَٰـأَيُّهَا الَّذِينَ ءَامَنُواْ ءَامِنُواْ بِاللَّهِ وَرَسُولِهِ وَالْكِتَٰـبِ الَّذِى نَزَّلَ عَلَى رَسُولِهِ وَالْكِتَٰـبِ الَّذِى أَنَزلَ مِن قَبْلُ
(O you who believe! Believe in Allah, and His Messenger, and the Book (the Qur'an) which He has revealed to the Messenger, and the Book which He sent own to those before (him)) (4:136),
وَلاَ تُجَٰـدِلُواْ أَهْلَ الْكِتَٰـبِ إِلاَّ بِالَّتِى هِىَ أَحْسَنُ إِلاَّ الَّذِينَ ظَلَمُواْ مِنْهُمْ وَقُولُواْ ءَامَنَّا بِالَّذِى أُنزِلَ إِلَيْنَا وَأُنزِلَ إِلَيْكُمْ وَإِلَٰـهُنَا وَإِلَٰـهُكُمْ وَاحِدٌ
(And argue not with the People of the Book, unless it be in (a way) that is better, except with such of them as do wrong; and say (to them): "We believe in that which has been revealed to us and revealed to you; our Ilah (God) and your Ilah (God) is One (i.e. Allah)') (29:46),
يَٰـأَيُّهَا الَّذِينَ ءَامَنُواْ ءَامِنُواْ بِاللَّهِ وَرَسُولِهِ وَالْكِتَٰـبِ الَّذِى نَزَّلَ عَلَى رَسُولِهِ وَالْكِتَٰـبِ الَّذِى أَنَزلَ مِن قَبْلُ
(O you who have been given the Book (Jews and Christians)! Believe in what We have revealed (to Muhammad ) confirming what is (already) with you) (4:47), and,
قُلْ يَٰـأَهْلَ الْكِتَٰـبِ لَسْتُمْ عَلَى شَىْءٍ حَتَّى تُقِيمُواْ التَّوْرَاةَ وَالإِنجِيلَ وَمَآ أُنزِلَ إِلَيْكُمْ مِّن رَّبِّكُمْ
(Say (O Muhammad ): "O People of the Book (Jews and Christians)! You have nothing until you act according to the Tawrah (Torah), the Injil (Gospel), and what has (now) been revealed to you from your Lord (the Qur'an).") (5:68).
Also, Allah the Exalted described the believers;
ءَامَنَ الرَّسُولُ بِمَآ أُنزِلَ إِلَيْهِ مِن رَّبِّهِ وَالْمُؤْمِنُونَ كُلٌّ ءَامَنَ بِاللَّهِ وَمَلَٰـئِكَتِهِ وَكُتُبِهِ وَرُسُلِهِ لاَ نُفَرِّقُ بَيْنَ أَحَدٍ مِّن رُّسُلِهِ
(The Messenger (Muhammad ) believes in what has been revealed to him from his Lord, and (so do) the believers. Each one believes in Allah, His Angels, His Books, and His Messengers. (They say,) "We make no distinction between any of His Messengers") (2: 285), and,
وَالَّذِينَ ءَامَنُواْ بِاللَّهِ وَرُسُلِهِ وَلَمْ يُفَرِّقُواْ بَيْنَ أَحَدٍ مِّنْهُمْ
(And those who believe in Allah and His Messengers and make no distinction between any of them (Messengers)) (4:152),
This is a sample of the Ayat that indicate that the true believers all believe in Allah, His Messengers and His Books.
The faithful among the People of the Book, have a special significance here, since they believe in their Books and in all of the details related to that, so when such people embrace Islam and sincerely believe in the details of the religion, then they will get two rewards. As for the others, they can only believe in the previous religious teachings in a general way. For instance, the Prophet ﷺ stated,
«إِذَا حَدَثَكُمْ أَهْلُ الْكِتَابِ فَلَا تُكَذِّبُوهُمْ وَلَا تُصَدِقُوهُمْ وَلكِنْ قُولُوا: آمَنَّا بِالَّذِي أُنْزِلَ إِلَيْنَا وَأُنْزِلَ إِلَيْكُم»
(When the People of the Book narrate to you, neither reject nor affirm what they say. Rather, say, 'We believe in what was revealed to us and what was revealed to you.')
However, the faith that many Arabs have in the religion of Islam as it was revealed to Muhammad ﷺ might be more complete, encompassing and firmer than the faith of the People of the Book who embraced Islam. Therefore, if the believers in Islam among the People of the Book gain two rewards, other Muslims who have firmer Islamic faith might gain an equal reward that compares to the two the People of the Book gain (upon embracing Islam). And Allah knows best.