The difference between the mushrikeen and the kuffaar, and to which category do the Jews and Christians belong?

Firstly: 

The kaafir is the one who denies and conceals the truth. The
basic meaning of the word kufr in Arabic is concealment. Shirk
means devoting worship to anyone or anything other than Allaah. 

Kufr may take the form of denying and rejecting, but the
mushrik may also believe in Allaah. This is the difference between the
mushrik and the kaafir. 

Each word may also carry the meaning of the other, so the
word kufr may be used in the sense of shirk, and the word shirk may be used
in the sense of kufr. 

Al-Nawawi (may Allaah have mercy on him) said: 

Kufr and shirk may carry the same meaning, which is disbelief
in Allaah, may He be exalted, or they may be used separately, whereby shirk
refers to the worship of idols and other created beings, whilst also
acknowledging Allaah, as the kuffaar of Quraysh did, and kufr may have a
more general meaning than shirk. End quote. 

Sharh Saheeh Muslim. 2/71 

Shaykh ‘Abd al-‘Azeez
ibn Baaz (may Allaah have mercy on him) said: 

Kufr is denial and concealment of the truth, such as one who
denies that prayer is obligatory, or that zakaah is obligatory, or that
fasting Ramadaan is obligatory, or that doing Hajj when one is able to is
obligatory, or that honouring one’s parents is obligatory, and so on, or one
who denies that zina is haraam, or that drinking intoxicants is haraam, or
that disobeying one’s parents is haraam, and so on. 

Shirk is devoting acts of worship to something or someone
other than Allaah, such as one who seeks the help of the dead, those who are
absent, the jinn, idols, the stars, and so on, or who offers sacrifices to
them, or makes vows to them. A kaafir may be called a mushrik and a mushrik
may be called a kaafir, as Allaah says (interpretation of the meaning): 

“And whoever invokes (or
worships), besides Allaah, any other ilaah (god), of whom he has no proof;
then his reckoning is only with his Lord. Surely, Al‑Kaafiroon (the
disbelievers in Allaah and in the Oneness of Allaah, polytheists, pagans,
idolaters) will not be successful”
[al-Mu’minoon 23:117]

“Verily, whosoever sets up partners (in worship) with
Allaah, then Allaah has forbidden Paradise to him, and the Fire will be his
abode”

[al-Maa’idah 5:72]

And Allaah says in Soorat Faatir (interpretation of the
meaning): 

“Such is Allaah, your Lord; His is the kingdom. And those,
whom you invoke or call upon instead of Him, own not even a Qitmeer (the
thin membrane over the date stone).

14. If you invoke (or call upon) them, they hear not your
call; and if (in case) they were to hear, they could not grant it (your
request) to you. And on the Day of Resurrection, they will disown your
worshipping them. And none can inform you (O Muhammad) like Him Who is the
All‑Knower (of everything)”
[Faatir 35:13-14]

Their calling on anything other than Allaah is called shirk
in this soorah, but in Soorat al-Mu’minoon it is called kufr. 

Allaah says in Soorat al-Tawbah (interpretation of the
meaning): 

“They (the disbelievers, the Jews and the Christians) want
to extinguish Allaah’s Light (with which Muhammad has been sent — Islamic
Monotheism) with their mouths, but Allaah will not allow except that His
Light should be perfected even though the Kaafiroon (disbelievers) hate
(it).

33. It is He Who has sent His Messenger (Muhammad) with
guidance and the religion of truth (Islam), to make it superior over all
religions even though the Mushrikoon (polytheists, pagans, idolaters,
disbelievers in the Oneness of Allaah) hate (it)”

[al-Tawbah 9:32-33]

Here Allaah calls the kuffaar kuffaar, and He calls them
mushrikeen. This indicates that a kaafir may be called a mushrik and a
mushrik may be called a kaafir. There are many similar verses and
ahaadeeth. 

Another example is the hadeeth in which the Prophet
(peace and blessings of Allaah be upon him) said: “Between a man and shirk
and kufr there stands his giving up prayer.” Narrated by Muslim in his
Saheeh from Jaabir ibn ‘Abd-Allaah (may Allaah be pleased with him). And
the Prophet (peace and blessings of Allaah be upon him) said: “The
covenant that differentiates us from them is prayer; whoever gives it up is
a kaafir.” Narrated by Imam Ahmad, Abu Dawood, al-Tirmidhi, al-Nasaa’i and
Ibn Maajah with a saheeh isnaad from Buraydah ibn al-Husayb (may Allaah be
pleased with him). And Allaah is the Source of strength. End quote. 

Majmoo’ Fataawa al-Shaykh Ibn Baaz,
9/174, 175. 

The Shaykh (may Allaah have mercy on him) also said:  

It is also shirk to worship only something other than Allaah.
This is called shirk, and it is called kufr. Whoever turns away from Allaah
altogether and devotes his worship to something other than Allaah, such as
trees, rocks, idols, the jinn or some of the dead such as those whom they
call awliya’ (“saints”), and worship them, pray to them or fast for them,
and forget Allaah altogether, this is the worst form of kufr and shirk. We
ask Allaah to keep us safe and sound. Similarly those who deny the existence
of Allaah and say that there is no god, that life is only material, such as
the communists and atheists, are the most disbelieving and misguided of
people, and the worst in terms of shirk. We ask Allaah to keep us safe and
sound. The point is that people who hold these and similar beliefs are all
regarded as mushrikeen and kaafirs who disbelieve in Allaah. Some people,
out of ignorance, mistakenly regard calling upon the dead and seeking their
help as waseelah and think that it is permissible. This is a serious error,
because this action is one of the worst forms of shirk and associating
others with Allaah. Some ignorant people and mushrikeen call it waseelah,
but it is the religion of the mushrikeen whom Allaah criticized and
condemned. He sent the Messenger and revealed His Books to denounce it and
warn against it. End quote. 

Majmoo’ Fataawa al-Shaykh Ibn Baaz,
4/32, 33 

Secondly: 

The Jews and Christians
are both kaafirs and mushrikeen. They are kaafirs because they deny the
truth and reject it. And they are mushrikeen because they worship someone
other than Allaah. 

Allaah says (interpretation of the meaning): 

“And the Jews say: ‘Uzair
(Ezra) is the son of Allaah, and the Christians say: Messiah is the son of
Allaah. That is their saying with their mouths, resembling the saying of
those who disbelieved aforetime. Allaah’s Curse be on them, how they are
deluded away from the truth!

31. They (Jews and Christians) took their rabbis and their
monks to be their lords besides Allaah (by obeying them in things which they
made lawful or unlawful according to their own desires without being ordered
by Allaah), and (they also took as their Lord) Messiah, son of Maryam
(Mary), while they (Jews and Christians) were commanded [in the Tawraat
(Torah) and the Injeel (Gospel)] to worship none but One Ilaah (God —
Allaah) Laa ilaaha illa Huwa (none has the right to be worshipped but He).
Praise and glory be to Him (far above is He) from having the partners they
associate (with Him)”

[al-Tawbah 9:30, 31]

Here they are described as mushrikeen. In Soorat al-Bayyinah
they are described as kaafirs, as Allaah says (interpretation of the
meaning): 

“Those who disbelieve from among the people of the
Scripture (Jews and Christians) and Al‑Mushrikoon, were not going to leave
(their disbelief) until there came to them clear evidence”

[al-Bayyinah 98:1]

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him)
said, refuting those who say that the word mushrikeen cannot be applied to
the People of the Book: 

It is most likely that the people of the Book are included
among the mushrikeen, men and women alike, when this word is used in general
terms. because the kuffaar are undoubtedly mushrikeen. Hence they are
forbidden to enter al-Masjid al-Haraam, because Allaah says (interpretation
of the meaning): 

“O you who believe (in Allaah’s Oneness and in His
Messenger Muhammad)! Verily, the Mushrikoon (polytheists, pagans, idolaters,
disbelievers in the Oneness of Allaah, and in the Message of Muhammad) are
Najasun (impure). So let them not come near Al‑Masjid Al-Haraam (at Makkah)
after this year”

[al-Tawbah 9:28]

If the People of the
Book did not come under the general heading of mushrikeen, then this verse
would not apply to them, and Allaah would not have referred to the beliefs
of the Jews and Christians in Soorat Baraa’ah (al-Tawbah) where He says
(interpretation of the meaning): 

“they (Jews and Christians) were commanded [in the Tawraat
(Torah) and the Injeel (Gospel)] to worship none but One Ilaah (God —
Allaah) Laa ilaaha illa Huwa (none has the right to be worshipped but He).
Praise and glory be to Him (far above is He) from having the partners they
associate (with Him)”

[al-Tawbah 9:31]

So they are all
described as mushrikeen, because the Jews said that ‘Uzayr is the son of God
and the Christians said that the Messiah is the son of God; and because they
took their priests and rabbis as lords instead of Allaah. All of this is the
worst form of shirk. And there are many similar verses. End quote. 

Majmoo’ Fataawa al-Shaykh Ibn Baaz,
4/274 

And Allaah knows best.