Algae and green plants might look similar in some ways but they are no more closely related to each other than they are to humans. The key similarity, the fact that they both perform photosynthesis, comes from a remote shared ancestor. Plants and algae have fundamental differences in structure, habitat and behaviour.
Algae and green plants, along with animals and fungi, do belong to the same domain, Eukarya, as distinguished from the two bacterial domains. This, however, is where the relationship ends. Eukarya contains four kingdoms: plants, fungi, animals and protoctists. Green plants belong to the kingdom Plantae, distinguished by, among other things, cell walls, multicellularity and immobility. Algae belong to Protoctista or Protista, a diverse group that includes, essentially, all the Eukaryotes that aren't animals, plants or fungi, including amoebas and many planktonic organisms. There are several groups of algae, and they aren't particularly closely related.
Structure and Development
Plants probably evolved from one group of green algal ancestors but have since become highly specialised with distinctive characteristics not shared with algae. Plants develop from an embryo, a characteristic they share with animals. Algae don't. Plants possess vascular connective tissue, lacking in algae. All plants are multicellular while most species of algae are single-celled organisms. Multicellular algae forms -- seaweeds -- do exist but they are in the minority.
Feeding and Movement
Green plants and many species of algae create their own food from sunlight by photosynthesis. However, some types of algae are partially or primarily hetereotrophs, they obtain their nutrients by eating other organisms or organic material. Many algae can swim, usually with the aid of one or more flagella, thin, hairlike structures. Plants are exclusively sessile. Plants have a limited range of movement, for example, they can lean towards a light source and some carnivorous plants have an automatic snap mechanism on leaves, but they do not have true mobility.
The vast majority of algae species are aquatic with a few adapted to extremely damp terrestrial environments. Most plants on the other hand are confined, and very well adapted, to land. A few plants have become secondarily adapted to aquatic environments in the manner of marine mammals. Their ancestors were terrestrial.