The difference between Kufr and Shirk

Imam Ibn Baaz, may Allah have mercy upon him answered a question:
"Disbelief is denying the truth and concealing it, such as one who denies the obligation of prayer, or the obligation of zakat, or the obligation of fasting in Ramadan, or the obligation of Hajj if one is able, or the obligation of honoring one’s parents, and so on..

Such as one who denies the prohibition of fornication, or the prohibition of drinking intoxicants, or the prohibition of disobedience to parents, or the likes of this.

As for Shirk, it is: directing some worship to other than Allah, such as one who beseeches help from the dead, the absent, the jinn, idols, the stars, and the like, or sacrifices to them or makes vows to them.

The Kaafir is called a Mushrik, and the Mushrik is called a Kaafir, as Allah the most high said:

(وَمَن يَدْعُ مَعَ ٱللَّهِ إِلَٰهًا ءَاخَرَ لَا بُرْهَٰنَ لَهُۥ بِهِۦ فَإِنَّمَا حِسَابُهُۥ عِندَ رَبِّهِۦٓۚ إِنَّهُۥ لَا يُفْلِحُ ٱلْكَٰفِرُونَ)
"And whoever invokes besides Allah another deity for which he has no proof -  then his account is only with his Lord. Indeed, the disbelievers will not succeed".
[Al-Mu'minun:117]

And the exalted said:

(إِنَّهُۥ مَن يُشْرِكْ بِٱللَّهِ فَقَدْ حَرَّمَ ٱللَّهُ عَلَيْهِ ٱلْجَنَّةَ وَمَأْوَىٰهُ ٱلنَّارُۖ)
"Indeed, he who associates partners with Allah - Allah has forbidden him Paradise, and his refuge is the Fire".
[Al-Ma'idah:72]

And the mighty and majestic said in Surah Faatir:

(وَٱلَّذِينَ تَدْعُونَ مِن دُونِهِۦ مَا يَمْلِكُونَ مِن قِطْمِيرٍ ۝ إِن تَدْعُوهُمْ لَا يَسْمَعُوا۟ دُعَآءَكُمْ وَلَوْ سَمِعُوا۟ مَا ٱسْتَجَابُوا۟ لَكُمْۖ وَيَوْمَ ٱلْقِيَٰمَةِ يَكْفُرُونَ بِشِرْكِكُمْۚ وَلَا يُنَبِّئُكَ مِثْلُ خَبِيرٍ)
"And those whom you invoke other than Him do not possess [as much as] the membrane of a date seed. If you invoke them, they do not hear your supplication; and if they heard, they would not respond to you. And on the Day of Resurrection they will deny your association. And none can inform you like [one] Acquainted [with all matters]".
[Fatir:13-14]

So he called their supplication to other than Allah Shirk in this Surah, and in Surah Al-Muminoon, He called it Kufr.

(يُرِيدُونَ أَن يُطْفِـُٔوا۟ نُورَ ٱللَّهِ بِأَفْوَٰهِهِمْ وَيَأْبَى ٱللَّهُ إِلَّآ أَن يُتِمَّ نُورَهُۥ وَلَوْ كَرِهَ ٱلْكَٰفِرُونَ ۝ هُوَ ٱلَّذِىٓ أَرْسَلَ رَسُولَهُۥ بِٱلْهُدَىٰ وَدِينِ ٱلْحَقِّ لِيُظْهِرَهُۥ عَلَى ٱلدِّينِ كُلِّهِۦ وَلَوْ كَرِهَ ٱلْمُشْرِكُونَ)
"They want to extinguish the light of Allah with their mouths, but Allah refuses except to perfect His light, although the disbelievers dislike it.
It is He who has sent His Messenger with guidance and the religion of truth to manifest it over all religion, although they who associate others with Allah dislike it.
[At-Taubah:32-33]

So He called the those that disbelief in Him Kufaar, and He called them Mushriks.

This points to that the Kaafir is called a Mushrik, and the Mushrik is called a Kaafir, and there are many verses and hadiths about that.
From it is the saying of the Prophet ﷺ: "Between a man and Shirk and Kufr is the abandonment of the prayer".
Reported by Muslim in his Sahih on the authority of Jabir bin Abdullah, may Allah be pleased with them both.

And his ﷺ saying: "The covenant between us and them is the prayer, so whoever abandons it has disbelieved". Reported by Imam Ahmad, Abu Dawud, Tirmidhi, Nasaa’i and Ibn Majah with an authentic chain of transmission on the authority of Buraidah bin Al-Hasib, may Allah be pleased with him, and Allah is the Grantor of success". End of speech.

Imam Ibn Baaz, may Allah have mercy upon him.

Question:
Are Shirk and Kufr synonymous, or is it the meanings are different when they are mentioned together?

Answer:
"They are synonymous generally speaking. And Kufr could be used upon rejection (Juhuud), and Shirk upon directing worship to other than Allah.

Every Kaafir is a Mushrik, and every Mushrik is a Kaafir.

But if it was due to rejection usually it's called Kufr, and if it is due to supplicating to other than Allah and or beseeching aid from other than Allah, it is called Shirk". End of speech.