Plants and animals generally differ in their lifestyles and the compositions of their bodies, but are fairly similar in terms of the chemical reactions and processes that allow them to stay alive and reproduce.
Plants are generally non-mobile; early in their evolution, they adopted a defensive strategy, sacrificing mobility for more conservative traits like thick cell walls. Animals took an offensive strategy, seeking out their food by moving to it. This led to a series of secondary adaptations; for example, animals are capable of higher rates of metabolism to support sudden bursts of movement, whereas plants have more of their bodies devoted to storage (such as in the form of large vacuoles) because they don't always know when they will encounter a supply of food or water.
Plants and animals have similar means of transporting materials within themselves, but the mechanism is slightly different. For example, both utilize a series of small tubes or capillaries, but animals tend to have a heart or some kind of pumping and pressure-maintaining organ, whereas plants utilize a more passive form of transport.
One of the most significant differences is reproductive strategies. Plants and some sessile (nonmoving) animals, such as coral and fish, still utilize sexual reproduction but don't actually engage in intercourse; instead they spread their gametes using water, air, or some other form of transport that isn't actually controlled by the organism, and essentially leave the fertilization and development of the offspring to chance. Unlike many "higher" animals, plants do not care for their young (although what we consider to be "care" also depends on having a nervous system, which plants lack).
This is another source of differences; while plants and animals both utilize specialized tissues to perform specific jobs within the body, plants lack many of the tissues that animals have, or they have analogues which do not function in exactly the same way, such as blood and sap. Plants are not conscious, cannot learn, and to the best of our knowledge, they are incapable of feeling pain.