The difference between algae and fungi is one of your basic questions in your life sciences classes in middle school. To just about anybody that can see, we can say that fungus are poisonous mushrooms that grow on your trees, your front yard, back yard, and so forth. And we can say that algae is the slimy green plant matter in the bottom of your home aquarium, your pool if it isn't chlorinated, or the bottom of the lake or sea. .
Fungi is plural for fungus. There are different types of fungi such as yeasts, molds, and mushrooms. They decompose and breakdown organic matter much faster. With fungi, it tends to be parasitic symbiosis meaning they feed off of other organisms aborbing nutrients into their cells.
In short, they decompose organic matter. Unlike algae which is plant matter, fungi do not use photosynthesis which is feeding off of sunlight. Instead, they feed off of the carbon of other organisms for metabolism. All living things have a carbon base. Humans are carbon based, dogs are carbon based, insects are carbon based, plants are carbon based. Water isn't considered a living thing because there's no carbon base.
If you leave a piece of fruit out in the open at room temperature, it's going to look all nasty after awhile. Because the fungus is going to surround it and form a mold. From there, the mold is going to eat it up.
In short, fungi are not considered plants. They are organisms that play in an important role of decomposures and break organic matter back to the earth. There are edible type of fungi such as Portobello mushrooms, oyster mushrooms, and straw mushrooms. Bleu cheese is a result of fungi and I do enjoy eating bleu cheese. However, there are numerous fungi that are toxis and fatal to humans. Meaning, if you see a mushroom in an uncontrolled environment, don't eat it.
Algae are different than fungi as they live in aquatic environments. Meaning, they exist in the water. They capture light through photosynthesis like plants do. As a result, inorganic becomes organic. Fungi can exist out of the water. Algae isn't parasitic unlike fungi is. Like fungi there are numerous forms of algae. Most important thing about algae is that its the ancestor of the first plants that existed on earth.
It's the primordial soup of the plants. In terms of symbiosis, it's mutual symbiosis in which the plant benefits and the algae benefits. Also algae doesn't decompose other organic matter. As humans, we use algae for fertilizer and food. For example, we eat seaweed which is a form of algae. I eat sushi and some of the rolls I order are wrapped in sushi.
Fungi and algae have their different uses amongst humans which is another way of distinguishing the differences. However, fungi and algae can merge together and form lichens.