What problems would have occurred in Rome if the city that large did not have an adequate sewage system?

Sewage systems have become a vital part in city planning to promote the health of the public and maintenance of public spaces.

The biggest benefit Roman society reaped from having an extensive sewage system was disease prevention. Humans invariably produce bodily waste throughout their lifetime, and this can be managed in a number of ways. Historically, latrines may have been dug on the outskirts of a settlement, or large pits were used as all-purpose waste...

Sewage systems have become a vital part in city planning to promote the health of the public and maintenance of public spaces.

The biggest benefit Roman society reaped from having an extensive sewage system was disease prevention. Humans invariably produce bodily waste throughout their lifetime, and this can be managed in a number of ways. Historically, latrines may have been dug on the outskirts of a settlement, or large pits were used as all-purpose waste disposal, including poo. In a large city like Rome, it became more difficult to manage human waste due to crowding. With lots of people living together in the same space, it is necessary to have a means of transporting human waste away from the living areas. Otherwise, waste would build up in fetid puddles and breed disease- not to mention the smell! Without a sewage removal system, Rome might have been quickly buried under all of the food scraps and bodily waste produced by the massive population. 

Contamination has been a big problem throughout the towns and cities of history, largely due to situations where people did not have dedicated means of removing human waste from their living spaces. Contamination of water and food re-introduced bacteria to the body and could easily spread disease like typhus.

Not only were the sewer systems beneficial for removing human waste and wastewater, they helped to drain out fallen rainwater and prevent flooding of the city spaces.