History and theology are interconnected so if you know about major historical differences, then you have an idea about some of the theological differences too.
On the surface, the divide is about who should've been the immediate (and later) successors to the Prophet (pbuh) invoking the question of merits and dimerits of Sahaba, but on a deeper level, it's about why all of that even matters!
I mean why it matters whether Ali or Abu Bakr must have been the successor? Ali got it on the fourth round anyway!
After showing that it was how Prophet (pbuh) and Allah had ordained it, Shia may go one (or many) level(s) deeper: The Prophet's ministry was an unfinished one. It was to be carried on by other God-aided individuals who would have brought the ministry into full fruition -- individuals who themselves were the perfect fruits of the Prophetic religion.
This usually already raises eyebrows for Sunnis and hence requires some further explanation:
Note that I said the ministry rather than the religion was unfinished. For the religion of Islam was finished when it was conceived in its primordial form in the first night of Qadr when Divine knowledge was granted to the Prophet in its totality, -- which was to be temporally particularized through the gradual revelation of the textual Quran in reference to various problems that Muslims or the Prophet faced during their struggles during their limited lifetime.
But rather the ministry, i.e. conveying Islam in its full depth was an unfinished task, a task to be carried on by other Divine-aided holy men, the Shia Imams until Islam was established in its pristine form globally.
All the special characteristics that the Shia attribute to their Imams, based on their own unique hadith sources, are requisites of this paramount religious task.
The Imams inherit the full knowledge of religion from the Prophet. It is important to highlight that this knowledge is not limited to knowledge that is taught and learned, but knowledge that is enabled by their access to supernatural realities, i.e. the Malakut, the Holy Spirit, the Protected Tablet, the Sidrat al-Muntaha, etc, i.e. the same sources that the Prophet himself was in contact with and derived his knowledge from. And with access to these supernatural realities come related supernatural abilities, i.e. kiramat or miracles.
This is really the core of Shia belief which is esoteric in nature. All else is secondary. And of course this is the most unfathomable part in Shia doctrine for Sunnis, if only in kinda masking the difference between the Prophet and the Imams or that how there's little trace of these (alleged) truths in Sunni traditions, and this would be the beginning of the debate.
But in short the Shia position is defended first by their distinction of the Prophet with revelation of Sharia and the Imams with its explication as well as by reference to the fact that the Imams owe their spiritual station directly or indirectly to initiation by Prophet Muhammad (pbuh) himself. The relative absence of clear and widespread reference to this doctrine in Sunni sources is explained by reference to the fact that Shia Imams were politically isolated making their doctrine inaccessible with mainstream Sunni scholars who canonized Sunni theology and law in the second and third Hijri century, and more fundamentally by reference to how these realities were hardly comprehensible for many (most) companions of the Prophet (who were just out of the state of Jahiliyya) during his own lifetime in the first place!
The Shia belief that the Prophetic ministry was unfinished is corroborated by the emergence of deep and numerous divisions over many key theological and legal questions after the passing of the Holy Prophet as well political conflicts and corruption. Either it is that the Prophet left behind a faulty controversial religion as his legacy with his close companions getting into conflict immediately after his death anticipating 14-odd centuries of religious conflict, or he did have special plans for the Ummah after his death, but personal interests of some individuals got in the way of fulfilling his advises about his proper spiritual and political succession that was necessary to shore up religion of Islam by continued elaboration of its generalities and clarification of its ambiguities and sensible outplay of its spirit -- a task no mean feat to be left to fallible companions.
You can further contemplate the various practical implications of this theology for yourself, like its politically revolutionary and troubled presence and the confusion and anger that outside encounter with its esoteric aspect has always provoked. So I suffice to this.
MA Islamic Philosophy and Theology, with focus on Shia Mysticism.