Difference between Restricted and Unrestricted Takbir in Dhul-Hijjah
Question

What is the difference between restricted and unrestricted takbir during the ten blessed days of Dhul-Hijjah? When does the takbir start? And what are its different formulas?
Praise be to Allah.
Firstly:
The virtues of Takbir in the first ten days of Dhul-Hijjah
The first ten days of Dhul-Hijjah are venerated days. Allah swore by them in His Book, and swearing by a thing is indicative of its importance and great benefit. Allah says (interpretation of the meaning):
“By the dawn; By the ten nights (i.e. the first ten days of the month of Dhul-Hijjah)” [Al-Fajr 89:1]
Ibn ‘Abbas, Ibn al-Zubayr, Mujahid and others among the earlier and later generations said: This refers to the ten days of Dhu’l-Hijjah. Ibn Kathir said: This is the correct view. (Tafsir Ibn Kathir, 8/413)
 Good deeds during these days are beloved to Allah, because the Prophet (peace and blessings of Allah be upon him) said: “There are no days in which righteous deeds are more beloved to Allah than these ten days.” They said, “O Messenger of Allah, not even jihad for the sake of Allah?” The Messenger of Allah (peace and blessings of Allah be upon him) said, “Not even jihad for the sake of Allah, except in the case of a man who went out to fight, giving himself and his wealth up for the cause, and came back with nothing.” Narrated by al-Bukhari, 969; al-Tirmidhi, 757. This version was narrated by al-Tirmidhi and classed as sahih by al-Albani in Sahih al-Tirmidhi, 605. 
One of the righteous deeds during these days is to remember Allah by reciting takbir (saying “Allahu akbar”) and tahlil (saying “La ilaha ill-Allah”), because of the following evidence. 

Allah says (interpretation of the meaning): 

“That they may witness [i.e., attend] benefits for themselves and mention the name of Allāh on known [i.e., specific] days over what He has provided for them of [sacrificial] animals. So eat of them and feed the miserable and poor.” [Al-Hajj 22:28]
The “appointed days” are the first ten days of Dhu’l-Hijjah. 

Allah says (interpretation of the meaning): 

“And remember Allah during the appointed Days” [Al-Baqarah 2:203]
These are the days of al-Tashriq, i.e. the 11th, 12th and 13th of Dhul-Hijjah. 

The Prophet (peace and blessings of Allah be upon him) said: “The days of tashriq are the days of eating, drinking and remembering Allah.” (Narrated by Muslim, 1141)

Secondly: 
Formulas of Takbir in the first ten days of Dhul-Hijjah
The scholars have different views on the formulas of Takbir. 
1- The first view is that it is: 
Allahu akbar, Allahu akbar, la ilaha ill-Allah, Allahu akbar, Allahu akbar, wa lillahi’l-hamd (Allah is Most Great, Allah is Most Great, there is no god but Allah, Allah is Most Great, Allah is Most Great, and to Allah be praise). 
2- The second view is that it is: 
Allahu akbar, Allahu akbar, Allahu akbar, la ilaha ill-Allah, Allahu akbar, Allahu akbar, Allahu akbar, wa lillahi’l-hamd (Allah is Most Great, Allah is Most Great, Allah is Most Great, there is no god but Allah, Allah is Most Great, Allah is Most Great, Allah is Most Great, and to Allah be praise). 
3- The third view is that it is: 
Allahu akbar, Allahu akbar, Allahu akbar, la ilaha ill-Allah, Allahu akbar, Allahu akbar, wa lillahi’l-hamd (Allah is Most Great, Allah is Most Great, Allah is Most Great, there is no god but Allah, Allah is Most Great, Allah is Most Great, and to Allah be praise). 
The matter is broad in scope because there is no text narrated from the Prophet (peace and blessings of Allah be upon him)which specifies a particular form of Takbir. 
Thirdly:
 The time of Takbir in Dhul-Hijjah
The takbir is divided into two categories: 
1 – Unrestricted Takbir. This is the takbir that is not limited to a specific time, and it is Sunnah to say it all the time, morning and evening, before prayer and after prayer, at all times. 
2 – Restricted Takbir. This is the takbir that is limited to the time immediately after prayer.
 It is Sunnah to recite unrestricted takbir at any time during the first ten days of Dhul-Hijjah and all the days of Tashriq, starting from the beginning of the month of Dhul-Hijjah (i.e., from sunset on the last day of Dhul-Qi’dah) until the end of the days of Al-Tashriq, which is when the sun sets on the thirteenth day of Dhul-Hijjah).
With regard to restricted takbir, it starts from Fajr on the day of ‘Arafah and lasts until the sun sets on the last day of the days of Tashriq, in addition to the takbir that may be recited at any time. So when a person says the taslim at the end of the obligatory prayer, he should pray for forgiveness (by saying Astaghfir-Allah) three times, and say, “Allahumma anta al-salam wa minka al-salam, tabarakta ya dha’l-jalali wa’l-ikram (“O Allah, You are al-Salam [the One Who is free from all defects and deficiencies), and from You is all peace, blessed are You, O possessor of majesty and honour”), then start the takbir.
 This is for those who are not on Hajj. The pilgrims should start the takbir after prayer from Zhuhr on the Day of Sacrifice (the 10th of Dhul-Hijjah).
 And Allah knows best.
 Reference: 
Majmu’ Fatawa Ibn Baz (may Allah have mercy on him), 13/17; 
Al-Sharh al-Mumti’ by Ibn ‘Uthaymeen (may Allah have mercy on him), 5/220-224.