There is no considerable difference between an Islamic government and the Islamic Republic in essence, the only thing is that the term ‘Islamic government’ is descriptive of a form of government whose different systems and policies, namely political, legal and economical, all originate from Islamic sources and teachings. Meanwhile, the term ‘Islamic Republic’ highlights the ‘structure’ of an Islamic government which (in our time) is the best frame and platform for the fulfillment of those Islamic and religious teachings.
What is understood from the whole of Islamic texts and authentic sources regarding government is that Islam isn't insistent on a particular form and structure with certain characteristics. Instead of prescribing a specific form of government in conformity with its stable and non-changing precepts, Islam has introduced a generic and encompassing framework that can take on changes, developments and different forms within itself (whilst conforming with those stable precepts at the same time). Islam hasn’t totally left the people on their own so that they do whatever they desire, nor has it dictated a very strict and restrictive method of government that cannot be applied in all times and places. The term designated for the generic and encompassing framework that Islam has introduced is ‘Islamic government and system’.
Therefore, the relation between ‘Islamic government’ and ‘Islamic Republic’ is that of ‘content’ and ‘framework’. ‘Islamic government’ denotes the content and inner structure, while ‘Islamic Republic’ is denoting of the method and form of application.
From the very start of the Islamic revolution, Imam Khomeini’s view was that the method of government and constitution of the country was to be determined by the people and put to their vote so that no particular form of government was imposed on them. That is why a nationwide vote was held on the 12th of Farvardin of the year 1358 (Iranian calendar) and it was unanimously decided (98 percent of the votes) by the people that the form of government be an ‘Islamic Republic’.
Of course, it should be kept in mind that there is still a long way left till a true Islamic society and establishing an Islamic government through an Islamic republic is only a starting point for becoming a society who can set an example for others, centered on God’s commands.