Praise be to Allah.
the points concerning which the four imams differed are many, and it is not possible to comment on them in all topics of fiqh in an article such as this answer, although it is possible to find out about them from the books which deal with differences of opinion among the madhhabs, such as al-Mughni by Ibn Qudaamah, Bidaayat al-Mujtahid by Ibn Rushd, and similar books.
But it is possible here to shed light on some of the reasons for differences of opinion among the fuqaha’ in general terms. Some of the reasons for their differences of opinion, for example, include the following:
1.Contradiction of the shar‘ i evidence in the eyes of the mujtahid.
The fuqaha’ differed in their methods of evaluating which reports were more sound, or reconciling between different reports. The scholars of the Standing Committee said, explaining the reasons for the differences of opinion among the scholars:
… Contradiction of the evidence: the fuqaha’ had different points of view with regard to evaluating which reports were more sound, or reconciling between different reports. For example, with regard to the hadith which says that it is not allowed to offer any prayer after ‘Asr until the sun has set, and the hadith which says that the one who enters the mosque should not sit down until he offers two rak‘ahs to “greet the mosque” (tahiyat al-masjid). The scholars differed concerning the application of that to one who enters the mosque at a time when praying is disallowed. Some of them gave precedence to the hadiths which say that praying at that time is disallowed, and others gave precedence to the hadith about greeting the mosque, and each side had evidence to support choosing which hadith should be given precedence.
End quote from Fataawa al-Lajnah ad-Daa’imah, 1 (5/31)
2.differences related to the Arabic language,
such as the difference of opinion regarding the interpretation of words that may, linguistically, have two or more meanings, such as the verse in which Allah, may He be exalted, says (interpretation of the meaning): “And divorced women shall wait (as regards their marriage) for three menstrual periods [quroo’]” [al-Baqarah 2:228]. The word quroo’ (translated here as “menstrual periods”), has two meanings; it may refer linguistically to the end of the menses, whereupon the woman becomes pure, or to the beginning of the menses. This text indicates that divorced women should wait for three quroo’. It may be that what is meant is the end of the third menses, or it may be that what is meant the beginning of the third menses. This has been discussed previously in fatwa no. 170581. Ibn al-Qayyim (may Allah have mercy on him) said in as-Sawaa’iq al-Mursalah (2/565):
For example, the differences that may arise as a result of the word having more than one meaning, or being ambiguous, or something that may be interpreted either according to the apparent meaning when used in general terms, which is called the real meaning, or may be interpreted according to the apparent meaning when used in specific terms, which is called the metaphor, such as their difference of opinion as to what is meant by quroo’: does it refer to the beginning or end of the menses? Some understood it as referring to the beginning of menses and others understood it as referring to the end of menses. End quote.
3.Differences of opinion concerning evidence that is general and whether it is valid or not
For example: the deeds of the people of Madinah, as transmitted through reports. Imam Maalik (may Allah have mercy on him) paid a great deal of attention to the deeds of the people of Madinah, and he thought that their deeds constituted valid proof regarding the religion of Allah, may He be exalted, and that it was not permissible to go against their consensus. This has been discussed previously in fatwa no. 179514.
4.Differences due to varying levels of knowledge and understanding of shar‘i texts
The scholars of the Standing Committee said: As for the reasons for differences of opinion among the scholars, there are many, such as the fact that none of them had complete knowledge of everything, hence a scholar may be unaware of something of which others are aware, and he may understand from a text something that others do not understand from it, when he is not aware of any unambiguous proof.
End quote from Fataawa al-Lajnah ad-Daa’imah 1 (2/178)
5.Differences due to the Sunnah (i.e., evaluation of hadith)
Such as the difference of opinion as to whether a hadith is saheeh (sound) or da‘eef (weak). Ibn al-Qayyim (may Allah have mercy on him) said in as-Sawaa‘iq al-Mursalah fi’r-Radd ‘ala al-Jahamiyyah wa’l-Mu‘attilah (2/556), when explaining the reasons for the differences of opinion:
…The third reason: belief that a hadith is da‘eef at a time when others may have a different view of that hadith based on their own examination of it. So one of the scholars may think that a narrator is da‘eef (weak) and not to be trusted, whilst another thinks that he is trustworthy and qawiy (strong). The correct view may be that of the one who regards the narrator as da‘eef, because he is aware of a reason of which the one who regards the narrator as trustworthy was unaware. Or the correct view may be that of the other one because he knows that that reason does not undermine the reports or good character of that narrator. End quote.
Or some of them may stipulate some conditions for aahaad reports that were not stipulated by others. Ibn al-Qayyim (may Allah have mercy on him) said in as-Sawaa‘iq al-Mursalah (2/559):
… The fourth reason: stipulation by some of them of conditions with regard to aahaad reports that differ from those stipulated by others. For example, some of them stipulated that the narrator should be a faqeeh, if he narrates something that is contrary to analogy; some of them stipulated that the hadith should be widely narrated and known if it discusses an issue that is widespread among the people; some of them stipulated that the hadith should not include any addition to the text of the Qur’an, lest that lead to abrogation of a Qur’anic text as a result.
6.Differences with regard to usooli principles
Such as their difference of opinion with regard to what is meant by a specific text: is it to be understood according to the general meaning of the text or is it made specific by a later text? Such as their difference of opinion regarding texts that speak of commands and prohibitions (and the degree of obligation or prohibition), and their difference of opinion with regard to the apparent meaning of the wording and how it is to be understood, and so on.
For more information, please see the answer to question no. 128658
With regard to the issue of following a specific madhhab among the four, this has been discussed in detail, quoting the opinions of the scholars, in fatwas no. 21420 and 103339. Please see also fatwa no. 148057 for the ruling on following a madhhab.
And Allah knows best.