There is currently no difference, but there have been substantial differences in the past.
Since the Courts Act 1971 (specifically s.42), the City of London has had its own county court that is on the same basis as other county courts. It is called "The Mayor's and City of London Court" for historical reasons, as that was the name of the court that was the previous closest equivalent. The 1971 Act establishes that the new version of the court has the same jurisdiction as any other county court, thus taking away any special privileges which the old version had. It was already almost like a county court by virtue of the City of London (Courts) Act 1964. Moreover, since the Crime and Courts Act 2013, there are no longer separate county courts, but one enormous one that sits in several places. That includes the City of London location.
In criminal matters, the Crown Court (established under the same 1971 Act) is logically also one enormous court. When it sits in the City of London, it is known as the "Central Criminal Court" (Senior Courts Act 1981, s.8(3)), but is still the same court. It does not have any special status with respect to the City, being meant to handle cases from all of Greater London as well. Formerly, it had to operate a complex system of separate juries for City cases, but nowadays that is all uniform.
For magistrates' courts, the Courts Act 2003 created a single jurisdiction for all of England and Wales. That is divided into "local justice areas", one of which is the City of London - these replace the old "petty sessional divisions". There are no special rules for the City. Applying the general rules for a particular criminal case would result in it being dealt with in the place where the offence is alleged to have happened, or where the accused lives, or where the witnesses mostly live, or "where other cases raising similar issues are being dealt with" (s.30(5)(d)).
Perhaps soon, powers under the Judicial Review and Courts Act 2022 will be exercised to abolish local justice areas (s.45), enabling even more flexibility with the assignment of cases to courts. In particular, this is meant to make it easer to shift cases around depending on local workload, or on which court location is genuinely most convenient in all the circumstances. The Act also removes (ss. 46 and 47) the current special rules about the City of London court premises being provided by the City authorities, rather than by the Ministry of Justice which handles all the other ones.
Formerly, there have been plenty of law courts for the City, or its livery companies, or St Paul's Cathedral (the consistory court had criminal jurisdiction over clergy), or the "compters" (debtors' prisons). Some of these still exist in a ceremonial capacity but do not function as courts. For example, the aldermen of the City can meet as the "Court of Husting", which was formerly a functioning law court but nowadays has the sole purpose of enrolling certain deeds, and hasn't done that since 1978. There are still two Sherriffs, nominally associated with the Poultry Compter and Wood Street Compter, but their courts have not existed since 1867. Some of the City's officers have residual ceremonial connections with the Old Bailey or other institutions, but these have no practical impact on the work of the courts.