during the history of Jews different events took place and they were put on trial different times and in many of them they failed, they showed ingratitude in different occasions and as a result, they faced the consequences[1], so their history is a great one to take lessons from...
In the Qur'an, Jews are mentioned in different verses for some reasons,
1. they have a long history and different prophets were sent to them so it is conceivable that they have been mentioned more than any other tribe.
2. during the history of Jews different events took place and they were put on trial different times and in many of them they failed, they showed ingratitude in different occasions and as a result, they faced the consequences[1], so their history is a great one to take lessons from.
For example, they were chosen by God but they didn't appreciate this fact and lost this opportunity and got punished by God, we Muslims should take lesson and learn from their story, we must appreciate the blessing of Islam that we have been presented with because if we don't, we will be afflicted with the same punishments that they were before.
3.in some of the verses God addresses the Jews and talks to them, maybe it is because God wants to remind them their past, the fact that they were chosen once but they lost that opportunity with ingratitude, so that they don't make the same mistakes twice and face the consequences again.
Like this verse of the Qur'an,
O Children of Israel, remember My favor that I bestowed upon you and that I preferred you over the worlds.[2]
4.another problem with regard to Jews was that they had some beliefs that were totally wrong and they insisted on them and some of these beliefs may even have distracted Muslims, for example according to some verses of the Quran, they believed that only they would enter the paradise or that they were the children of Allah and His beloved ones. So we can see that in some verses of the Qur'an God condemns these wrong beliefs and corrects them, like this verse of the Qur'an,
And they say, "None will enter Paradise except one who is a Jew or a Christian." That is [merely] their wishful thinking, Say, "Produce your proof, if you should be truthful." Only he who surrenders to God with all his heart and also does good, will find his reward with his Lord, and will have no fear or regret.[3]
The Jews and the Christians say, ‘We are Allah’s children and His beloved ones.’ Say, ‘Then why does He punish you for your sins?’ No, you are humans from among His creatures. He forgives whomever He wishes, and punishes whomever He wishes, and to Allah belongs the kingdom of the heavens and the earth, and whatever is between them, and toward Him is the return.[4]
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[1] for one example you can refer to the chapter 5 verse from the verse 21 to 26.
[2] Holy Quran verse 47 of chapter 2. Al-Baqara
يا بَنِي إِسْرَائِيلَ اذْكُرُوا نِعْمَتِيَ الَّتِي أَنْعَمْتُ عَلَيْكُمْ وَأَنِّي فَضَّلْتُكُمْ عَلَى الْعَالَمِينَ
[3] Holy Quran verse 111,112 of chapter 2. Al-Baqara
وقَالُوا لَن يَدْخُلَ الْجَنَّةَ إِلَّا مَن كَانَ هُودًا أَوْ نَصَارَىٰ ۗ تِلْكَ أَمَانِيُّهُمْ ۗ قُلْ هَاتُوا بُرْهَانَكُمْ إِن كُنتُمْ صَادِقِينَ. بَلَىٰ مَنْ أَسْلَمَ وَجْهَهُ لِلَّهِ وَهُوَ مُحْسِنٌ فَلَهُ أَجْرُهُ عِندَ رَبِّهِ وَلَا خَوْفٌ عَلَيْهِمْ وَلَا هُمْ يَحْزَنُون
[4] [5:18]
وَقَالَتِ الْيَهُودُ وَالنَّصَارَىٰ نَحْنُ أَبْنَاءُ اللَّـهِ وَأَحِبَّاؤُهُ ۚ قُلْ فَلِمَ يُعَذِّبُكُم بِذُنُوبِكُم ۖ بَلْ أَنتُم بَشَرٌ مِّمَّنْ خَلَقَ ۚ يَغْفِرُ لِمَن يَشَاءُ وَيُعَذِّبُ مَن يَشَاءُ ۚ وَلِلَّـهِ مُلْكُ السَّمَاوَاتِ وَالْأَرْضِ وَمَا بَيْنَهُمَا ۖ وَإِلَيْهِ الْمَصِيرُ