This is past your time limit, but the idea of Arcologies (popularized by Paolo Soleri in his book: Arcology: City in the Image of Man) is for a very dense 3 dimensional construction where people are close to each other and all the goods and services they would want or need.
For earlier periods, vertical cities would have some disadvantages, including difficulty in accessing the vertical aspects of the cities (immature elevator technology), crowded streets and issues with parking and utility services (a lot of energy would be needed to pump water to the upper floors, and "grey" and "black" water racing down the pipes would have some interesting effects downstream. As well, food and other goods and services would have to be passed through crowded streets, and wastes passed back the same way. electrical infrastructure and heat in the form of coal or oil fired furnaces would also be hard pressed to keep up.
The "advantages" of this scenario would be that major engineering challenges would have to be met, so things like pumps and urban infrastructure would be refined to a much higher pitch earlier in history than we saw. As well, urban planning would take on a different cast, as much of the urgency would be to solve the problems of moving large amounts of goods in and out of the city as efficiently as possible. Different patterns of road building, more mass transit and train transport and even elevated and subterranean transport arteries would all be part of city building and planning. Mass communications would also be advanced in order to service the huge audiences framed into small areas.
This would also have changes in social and political institutions, perhaps making the rise of "mass political movements" like Fascism come earlier, since maipulating large groups of people would be much easier in urban settings with late 19th and early 20th century technology.
Since we see that in the "real world" people are not entirely keen on living this sort of lifestyle, we should also expect to see an outflow of people from the "vertical" city and towards outlying small towns or prototype suburbs (which began to develop in the 1930's, much at the same time that "vertical" technology was also maturing; an interesting coincidence).
So like everything else, "vertical" cities could be good (like some of Frank Lloyd Wrights visions) or bad (much like the city in the movie "Metropolis").