Social vs. Solitary
The most obvious characteristic that ants have that grasshoppers don't is that ants are social insects and grasshoppers are usually solitary. Ants live in colonies that can be made up of millions of individuals. These individuals defend the colony, forage for food, raise the young, and tend the queen(s).
Grasshoppers, however, fend for themselves and usually only come together to mate. Grasshoppers can sometimes form swarms that do terrible damage to crops. In this guise they're called locusts. A swarm can be made up of billions of locusts but there is no underlying social structure as there is in ant colonies.
Care of young
Ants tend eggs and raise larvae in their colony while grasshoppers do not care for their offspring. After mating the female grasshopper lays her eggs in the ground, then leaves. The nymph hatches, tunnels up through the ground, and is on its own.
Defense
Ants can bite and some species can sting. An ant's stings can be painful, but they're rarely fatal. Grasshoppers don't bite or sting but if handled some will vomit up "tobacco juice," which is the contents of their stomachs.
Life cycle
Ants mature through complete metamorphosis, which means the ant begins as a larva and must progress to a pupal stage before becoming an adult. Grasshoppers mature through incomplete metamorphosis, which means the nymph resembles an adult save wings and mature reproductive organs. It will molt four to six times before reaching adulthood.
Ants are a young(ish) species. Grasshoppers are considerably older. Ants arose during the Cretaceous period, around 110 million years ago. Grasshoppers arrived in the late Permian period, about 250 million years ago.
Cultivation of resources
Some species of ants make slaves out of other ants, grow fungus farms, and keep aphids and caterpillars as livestock. The aphids and caterpillars are prized by the ants for their honeydew. Honeypot ants eat till they're engorged with food and serve as a food source for their nestmates when food becomes scarce. Grasshoppers, on the other hand, don't keep slaves or fungus farms and they don't help feed other grasshoppers.