The short answer is that the difference in the content of hadith, too, stem from the same significant questions of Islamic political doctrine that originally divided the two sects.
For Shia scholars, Shia and Sunni hadith has no difference and are checked with the same methods of checking authenticity . If a hadith is known to be authentic, it is used, but if not, it is left, whether it is recorded by Shia or Sunni scholars. Shia scholars do not reject Sunni hadith books completely, but (say) every individual hadith (from Shia or Sunni) should be investigated for authenticity.
Shia scholars use authentic hadith from Sunni hadith book in their works and books. For example, the hadith from Sahih Muslim and other Sunni hadith books are used in Imam Khomeini books, and the books of many other Shia scholars.
Please note there are different methods of determining the health of a hadith and this makes some conflict in scholars about considering a hadith authentic or not. This conflict exists in Shia scholars. For example, a Shia scholar may consider a hadith and other inauthentic due to different methods of determining the health of a hadith and this may lead to different fatwa an a subject.
The main fundamental difference of Shia and Sunni is conflict in Imamat which is a very big conflict and this conflict makes big different in every aspect of Islam including hadith collections, Madhab, prayers, beliefs and everything else. There are tens of thousands of hadith that accepting them as authentic will in fact means accepting Imamat as a pillar of Islam and in a way are related to Imamat. So those kind of hadith are totally censored in main Sunni hadith books but still many of them can be found in many Sunni hadith books.
Sunni scholars are different about recording hadith related to Imamat and it depends on the relation of that scholar to King of his time and pressures of King of that time on scholars. In fact Imamat is opposite of Kingship. A King accepting Imamat in fact is as accepting himself as not qualified for governing Islamic society; so no King will never accept Imamat (if he accepts he is not a King anymore) and this caused Kings always controlling scholars and hadith books to not write anything related to Imamat.
In Shia view, a good hadith narrator for Sunni Kings and hadith recorders is who be silent about Imamat and Caliphate and support power of Sunni Caliphs and Kings even if that narrator is from Kharijites. And if a narrator oppose Sunni Caliphs and Kings, he should not be used as a narrator even if he is Imam Ali a.s. or Imam Sadiq a.s.; unless their hadith is not talking about Caliphate.
After Malik the golden age of hadith started and Sunni collections were written in this age.
This is some of views of Shia scholars regarding 6 main Sunni hadith collections:
1- Sahih Bukhari (year 194-256): the Sunni hadith recorders have had many exaggerations about this book while this book narrates hadith from Kharijates and Nasibi people. But from Imam Sadiq a.s. and next Imams who Bukhari was living at their time even one hadith is not narrated. His ancestors were Zoroastrian and first one of their family who converted to Islam was his 3th father Mughayrah. When Bukhari was child his father died and left a wealth that he used for travels for collecting hadith. Firstly he was respected by scholars of Nishabur but later due to his serious conflicts with Muhammad Ibn Yahya Nishaburi (the Grand Hadith Sheykh of Nishabur) he was forced to immigrate from his city.
2- Sahih Muslim (year 204-261): although this book has advantages that Bukhari book does not have but is in lower rank and popularity.
3- Sahih Abu Dawood: its ahadith are mostly about Sunni Fiqh.
4- Sahih Abu Isa Muhammad Ibn Isa Tarmazi: he was the first one who categories ahadith in 3 category of Sahih, Hasan, Zaif instead of 2 category.
5- Sahih Ahmad Ibn Shoaib Nesai.
The last 4 are called Sunan also because the terms considered for Sahih hadith by first two are not considered in these 4.
6- Sahih Muhammad Ibn Yazid Ibn Abdullan Ibn Majeh Ghazwini: some consider Sunan Darami higher than this book and consider it as 6th Sahih.