The difference between du’aa’ for asking and du`a for worship
Question

What is the difference between du`a for asking and du’aa’ for worship?.
Praise be to Allah.We use the word du`a (invocation) to refer to two things: 
1-
du`a for asking, which means seeking that which will benefit, or asking to ward off that which will cause harm, by asking Allah for that which will benefit in this world and in the Hereafter, and to ward off that which will cause harm in this world and in the Hereafter, 
Such as du`a for forgiveness, mercy, guidance, help, to attain Paradise, to be saved from Hell, for Allah to grant good things in this world and in the Hereafter and so on. 
2-
du`a for worship, which means that the person is worshipping Allah by any of the types of worship, in the heart or physically or financially, such as fearing Allah, loving Him, hoping for His mercy, putting one’s trust in Him, praying, fasting, doing Hajj, reading Qur’aan, reciting tasbeeh and dhikr, giving zakaah and charity, jihad for the sake of Allah, calling people to Allah, enjoining what is good and forbidding what is evil, and so on. 
Everyone who does any of these acts of worship is calling upon Allah. 
See: al-Qawl al-Mufeed (1/264) and Tasheeh al-du`a (p. 15-21). 
Usually when the word du`a appears in verses of the Qur’aan, it refers to both meanings, because they are interconnected. So everyone who asks of Allah verbally is worshipping Him, because du`a is worship, and every worshipper who prays to Allah, fasts or does Hajj is doing that hoping for reward from Allah and to attain Paradise and to be saved from punishment. 
Shaykh ‘Abd al-Rahmaan al-Sa’di (may Allah have mercy on him) said: 
Everything in the Qur’aan that enjoins du`a and forbids du`a to or calling upon anyone other than Allah, and praises those who say du`a or call upon Him, includes the du`a of asking and the du`a of worship. End quote. 
Al-Qawaa’id al-Hisaan (no. 51). 
One of the two types of du`a may be more apparent than the other in some verses. 
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said concerning the verse (interpretation of the meaning): “Invoke your Lord with humility and in secret. He likes not the aggressors. 56. And do not do mischief on the earth, after it has been set in order, and invoke Him with fear and hope. Surely, Allah’s Mercy is (ever) near unto the good‑doers” [Al-A’raaf 7:55-56]:  
These two verses include the etiquette of two types of du`a: the du`a of worship and the du`a of asking. du`a in the Qur’aan may refer to either of these types, or it may refer to both, because they are interconnected. du`a of asking means seeking that which will benefit the supplicant, and asking to be spared that which will harm him, and for it to be warded off. So he asks for benefit and for harm to be warded off, and that is du`a of asking. And the du`a that is offered with hope and fear is the du`a of worship. Therefore it is known that both types are interconnected. Every du`a of worship implies du`a of asking, and every du`a of asking implies du`a of worship. 
Based on this, the words (interpretation of the meaning): “And when My slaves ask you (O Muhammad صلى الله عليه وسلم) concerning Me, then (answer them), I am indeed near (to them by My Knowledge). I respond to the invocations of the supplicant when he calls on Me (without any mediator or intercessor)” [al-Baqarah 2:186] include both types of du`a, and it is in terms of both that the verse is to be understood. It was said: I will give to him when he asks of Me, and it was said: I will reward him if he worships Me. Both are interconnected. 
This does not come under the heading of using a common word in both of its meanings, or of using a word in a real and metaphorical sense, rather it is using it in its real sense which includes both things. 
Think about it, because it is a matter that is of great benefit, but it is rarely paid attention to. Most verses of the Qur’aan point to both meanings and more, and it is of this type. 
For example, Allah says (interpretation of the meaning):  
“Say (O Muhammad صلى الله عليه وسلم to the disbelievers): My Lord pays attention to you only because of your invocation to Him”
[Al-Furqaan 25:77]. 
i.e., because of your du`a to Him. And it was said: (it means) because of His call to you to worship Him.  
Based on that, what is meant is both types of du`a, of which is the du`a of worship is more apparent, i.e., He would not have paid attention to you were it not that you put your hope in Him. Worship of Him implies asking of Him, so both types are included. 
And Allah says (interpretation of the meaning):  
“And your Lord said: Invoke Me [i.e. believe in My Oneness (Islamic Monotheism) and ask Me for anything] I will respond to your (invocation)” [Ghaafir 40:60]. 
The word du`a here includes both types, although du`a of worship is more apparent, hence it is followed by the words (interpretation of the meaning): “Verily, those who scorn My worship [i.e. do not invoke Me, and do not believe in My Oneness, (Islamic Monotheism)] they will surely enter Hell in humiliation!”. So the word du`a in this verse may be interpreted in both senses. 
Al-Tirmidhi narrated that al-Nu’maan ibn Basheer (may Allah be pleased with him) said: I heard the Messenger of Allah (peace and blessings of Allah be upon him) say on the minbar: “du`a is worship.” Then he recited the verse (interpretation of the meaning): “And your Lord said: Invoke Me [i.e. believe in My Oneness (Islamic Monotheism) and ask Me for anything] I will respond to your (invocation)…”. Al-Tirmidhi said: a saheeh hasan hadeeth. 
With regard to the verses (interpretation of the meaning):  
“Verily, those on whom you call besides Allah, cannot create (even) a fly, even though they combine together for the purpose…” [Al-Hajj 22:73] 
“They (all those who worship others than Allah) invoke nothing but female deities besides Him (Allah) …” [Al-Nisa’ 4:117] 
“And those whom they used to invoke before (in this world) shall disappear from them…” [Fussilat 41:48]
everything that is mentioned in them is the du`a or invocation of the mushrikeen to their idols. What is meant is the du`a of worship which includes the du`a of asking, although the meaning of du`a of worship is more apparent. 
The words (interpretation of the meaning): “so invoke Him making your worship pure for Him Alone (by worshipping Him Alone, and none else” [Ghaafir 40:65] refer to the du`a of worship. What is meant is: Worship Him alone and make your worship sincerely for Him alone, and do not worship anyone else along with Him. 
With regard to the words of Ibraaheem (peace be upon him) in the verse (interpretation of the meaning): “ ‘Verily, my Lord is indeed the All‑Hearer of invocations’” [Ibraaheem 14:39], what is meant by hearing here is hearing in a specific sense, which is the hearing of response and acceptance, not hearing in the general sense, because Allah hears all that can be heard. As that is the case, du`a here means the du`a of worship and the du`a of requesting, and the Lord’s hearing of it means that He rewards the praise and answers the request, so He hears both. 
With regard to the words of Zakariya (peace be upon him) in the verse (interpretation of the meaning):  “ ‘and I have never been unblest in my invocation to You, O my Lord’” [Maryam 19:4], it was said that what is meant is the du`a of asking, meaning: You always answered me and never  rejected me or deprived me. So he is seeking to draw closer to Allah, may He be glorified and exalted, by means of His previous answers and kindness. This is clear in this case.  
With regard to the words of Allah (interpretation of the meaning):
“Say (O Muhammad): Invoke Allah or invoke the Most Gracious (Allah)…” [Al-Isra’ 17:110], 
The well known view is that this is the du`a of asking, and this is the reason for revelation. They said: The Prophet (peace and blessings of Allah be upon him) used to call upon his Lord, sometimes saying Ya Allah and sometimes saying Ya Rahmaan, so the mushrikeen thought that he was calling upon two gods, so Allah revealed this verse. 
With regard to the verse (interpretation of the meaning): “Verily, We used to invoke Him (Alone and none else) before. Verily, He is Al‑Barr (the Most Subtle, Kind, Courteous, and Generous), the Most Merciful” [al-Toor 52:28], this is the du`a of worship which includes asking with hope and fear. What it means is: we used to devote our worship sincerely to Him alone, and thus they deserved that Allah should save them from the punishment of Hell, not just because of asking, which is something that is shared in common between those who are saved and those who are not, because everyone who is in heaven and on earth asks for that. “never shall we call upon any ilaah (god) other than Him” [al-Kahf 18:14] i.e., we will never worship anyone other than Him. Similarly Allah says (interpretation of the meaning): “Will you call upon Ba‘l (a well-known idol of his nation whom they used to worship) and forsake the Best of creators?” [Al-Saffaat 37:125].  
With regard to the words (interpretation of the meaning): “And it will be said (to them): ‘Call upon your (so‑called) partners (of Allah)’, and they will call upon them…” [al-Qasas 28:64], this is du`a of asking. Allah will rebuke and humiliate them on the Day of Resurrection, by showing them that their partners (whom they ascribed to Allah) will not answer their du`as or supplications. But it does not mean worship them. It is similar to the verse in which Allah says (interpretation of the meaning): “And (remember) the Day He will say: ‘Call those (so‑called) partners of Mine whom you pretended.’ Then they will cry unto them, but they will not answer them” [Al-Kahf 18:52]. End quote.  
Majmoo’ Fatawa Ibn Taymiyah (15/10-14). 
See other examples in Badaa’i’ al-Fawaa’id by Ibn al-Qayyim (3/513-527). 
And Allah knows best.