Is It Prohibited to Delay Prayer?
Question

My question is about the time for prayer. The time begins when the Adhan is given, but when does the time end? Is there a difference between delaying prayer and missing the time for prayer? And what is the punishment for both?
Praise be to Allah.To find out about the times of prayer and when they begin and end, please see question no. 9940 . 
Difference between delaying prayer and missing the time for prayer
With regard to your question about the difference between delaying prayer and missing the time for prayer, the answer is as follows: 
Missing the time for prayer
Missing the time for prayer means delaying the prayer until the time is over and one has not prayed. This is a major sin, unless that is for a legitimate excuse such as sleeping or forgetting. 
It says in Al-Mawsu`ah Al-Fiqhiyyah, 10/8: “The jurists agreed that it is forbidden to delay prayer until the time for prayer is over, without a legitimate excuse.” 
Shaykh `Abd Al-`Aziz ibn Baz (may Allah have mercy on him) said: “With regard to the person who deliberately delays the prayer until after the time for it is over, or who sets his alarm clock late so that he will not get up on time, he is deliberately not praying, and he has committed a great evil according to all the scholars, but is this disbelief or not? 
Here there is a difference of opinion among the scholars: 
If he does not deny that prayer is obligatory, then the majority of scholars say that he is not committing major disbelief  in this case. 
Some of the scholars were of the view that this is major disbelief which puts him beyond the pale of Islam, because the Prophet (peace and blessings of Allah be upon him) said: “Between a man and shirk [associating others with Allah] and disbelief  there stands his giving up prayer.” (Narrated by Muslim, 82)
And the Prophet (peace and blessings of Allah be upon him) said: “The covenant that stands between us and them is prayer; whoever gives up prayer is a disbeliever.” (Narrated by Ahmad and the four authors of As-Sunan with an authentic Isnad [chain of narrators]) And there is other evidence to support this.
It was also narrated from the Companions of the Prophet (may Allah be pleased with them all). 
The great Tabi’i ‘Abdullah ibn Shaqiq Al-`Aqili (may Allah have mercy on him) said: “The Companions of the Messenger of Allah (peace and blessings of Allah be upon him) did not think that omitting anything constituted disbelief apart from prayer.” 
Is it prohibited to delay prayer?
With regard to delaying prayer , this may refer to two things: 
1.     Delaying the prayer until the time for that prayer is over. We have explained the ruling on this above.
2.     Delaying the prayer until the end of the time for that prayer. (See Al-Mawsu`ah Al-Fiqhiyyah, 10/6) 
It is permissible to offer a prayer at the end of its time, because of the Hadith narrated by Muslim (614) from Abu Musa Al-Ash’ari (may Allah be pleased with him), that someone came to the Messenger of Allah (peace and blessings of Allah be upon him) and asked him about the times of prayer… and the Prophet (peace and blessings of Allah be upon him) explained to him the beginning and end of the time for each prayer, and said, “The time is between these two.” 
But if delaying the prayer means that one will miss out on praying in congregation and will pray it alone at the end of its time, then it is prohibited because it means not praying in congregation, so long as one does not have an excuse for not praying in congregation. 
Is it Sunnah to delay `Isha’?
It is better to do the prayer at the beginning of its time, except for `Isha’ prayer , and also Dhuhr prayer when it is very hot; in these cases it is better to do them at the end of their allotted times. 
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said: 
“It is better to do the prayer at the time required in the Shari`ah. Hence the Prophet (peace and blessings of Allah be upon him) said, answering the one who asked him which deed is most beloved to Allah, “Prayer done on time .” (Narrated by Al-Bukhari, 527 and Muslim, 85) He did not say prayer at the beginning of its time. That is because the prayers include some which it is Sunnah to pray sooner and some which it is Sunnah to pray later. It is Sunnah to pray `Isha’ later, after one-third of the night has passed. Hence if a woman is at home and asks which is better for me, to pray `Isha’ when I hear the Adhan or to delay it until one-third of the night has passed? We say, it is better to delay it until one-third of the night has passed, because the Prophet (peace and blessings of Allah be upon him) delayed one night until they said, “O Messenger of Allah, the women and children have fallen asleep,” so he went out and led them in prayer, and said, “This would be its time, were that not too difficult for my Ummah.” So if a woman is in her house, it is better for her to delay it. 
Similarly, if it so happens that some men are travelling, and they ask whether they should pray `Isha’ sooner or later, we would tell them that it is better for them to delay it. 
Similarly if a group goes out for a trip and the time for Isha’ comes, is it better for them to pray `Isha’ sooner or later? We say that it is better for them to delay it if that will not cause them any hardship. 
In the case of the other prayers it is better to pray them sooner unless there is a reason for doing otherwise. So Fajr should be prayed sooner , Dhuhr should be prayed sooner , `Asr should be prayed sooner and Maghrib should be prayed sooner, unless there is a reason for doing otherwise.  
Reasons for delaying prayer 
Such reasons include the following: 
·        If it is intensely hot, then it is better to delay Zuhr prayer until it cools down a little, i.e., until it is nearly time for `Asr, because it cools down a little when it is nearly time for `Asr. When it is intensely hot, it is better to pray when it is cooler, because the Prophet (peace and blessings of Allah be upon him) said: “When it is intensely hot, then wait until it cools down before you pray, for intense heat is from the breeze of Hell.” (Narrated by Al-Bukhari, 537 and Muslim, 615)
The Prophet (peace and blessings of Allah be upon him) was on a journey and Bilal (may Allah be pleased with him) stood up to give the call to prayer, and he said, “Wait until it gets cooler.” Then [Bilal] stood up to give the call to prayer, and [the Prophet (peace and blessings of Allah be upon him)] said, “Wait until it gets cooler.” Then [Bilal] stood up to give the call to prayer, and [the Prophet (peace and blessings of Allah be upon him)] gave him permission to do so. (Narrated by Al-Bukhari, 629 and Muslim, 616) 
·        Another reason is if it will be possible to pray in congregation at the end of the time but not at the beginning, in which case delay is preferable. For example, if a man is outside the city when the time for prayer comes, and he knows that he will reach the city and catch up with the congregation at the end of the time for prayer, is it better for him to pray when the time for prayer comes, or to delay the prayer until he reaches the congregation? 
We say that it is better for him to delay the prayer until he catches up with the congregation; we say that it is even obligatory for him to delay in this case, so that he can catch up with the congregation.” (Fatawa Arkan Al-Islam, p. 287)
And Allah knows best.