Freshwater fish are considered to be more robust to water conditions. This is often why people start off with a freshwater tank. The fish have adapted through the ages to living in water that might change rapidly if there is a flood or a drought. Saltwater fish are not this adaptable because they have lived in the sea that doesn't change much in terms of water quality. The volume of water in the sea is so large that any chemical disturbances are diluted to be negligible. Of course this volume of water is not the case in a small sea water fish tank so the responsibility for maintaining the water quality is down to the owner.
One of the obvious differences in saltwater aquariums to freshwater is that there is salt in the water. Due to the process of osmosis water flows from areas of low salinity to high salinity. In the case of saltwater fish there is more salt in the water than in the fish. Thus the fish will lose water to it's environment and constantly has to drink water to preserve the balance and stay healthy. Freshwater fish are exactly the opposite. They have more salt in the body than in the water so water passes through the cells of the fish by osmosis.
Thus the fish are very different. Freshwater fish need to rid the body of excess water and saltwater fish need to rid the body of excess amounts of salt in order to be in harmony with the environment. Saltwater fish need to work harder to maintain harmony within the environment. They are also more susceptible to changes in the chemistry of the water because they have to drink more of it.
Thus the most important part of keeping saltwater fish is maintaining the specific gravity of the fish tank water. This is the level of salt in the water. You need to understand the specific gravity that the species of saltwater fish you are keeping is comfortable in. Then you need to keep the water at this gravity.
Other differences that are determined by the environment that the animals live in are the types of foods that they eat. This will vary from species to species but in general the saltwater variety will be more particular about what they eat.
Keeping a saltwater aquarium is more complex than a freshwater tank because there is more responsibility on the fish tank owner to ensure the water quality is correct. However the benefits of keeping a saltwater tank are more diverse environments and species to care for.