Rome functioned in many ways similarly to how it had in the past. The Emperor and Senate had never had direct control over the city, but rather were responsible for appointing those posts that exercised that control. Now, the Emperor still provided funds for infrastructure and building projects. but in terms of the day to day running, the police, the fire brigades, the aqueducts, the sewers, etc, all had magistrates (elected in the Republic, but generally appointed in the Imperial period) whose responsibility it was to maintain specific aspects of the city infrastructure, oversee games, and the grain dole.

In the sense of your question about the distinction between the city and Empire, the Senate and other legislative bodies were the government of the city, and the Emperor, at least during the Principate, was merely the 'first citizen' of the city. And that city and her armies controlled a vast Empire, which was one of the main reasons for the transition to an Empire from the Republic in the first place; imagine trying to run the United States with just the New York city council as your government. It simply wasn't designed to cope and it couldn't. What this meant in practice is that although the government for the city didn't change much in the Imperial period, the government for the provinces of that city evolved. While officially nothing changed in the inherent structure, the de facto control of the legions by the emperor (almost entirely at first, and then entirely within a generation or two) meant that the de jure authority of the city's legislative bodies over the rest of the territory outside of the city was moot.