Algae are a group of lower photosynthetic aquatic plants.  The body of algae is called thallus. It may be unicellular or multicellular. Generally the algae are classified into three groups based on the presence of photosynthetic and accessory pigments. They are  brown,  red and green algaeThe branch of biology that deals with algal studies is called Phycology.

Fungi (sing. Fungus) are group of heterotrophic , spore bearing and non vascular organisms without chlorophyll.  Fungi are quite diverse in their organization ; some are one celled like yeast and others are very complex like mushrooms. Mycology is the branch of biology which deals with the study of fungi. Classification of Fungi
Algae vs Fungi
Algae vs Fungi
Algae Fungi
Algae They are mostly aquatic, found in fresh as well as marine water.
Most of the fungi are terrestrial grow mostly on land where organic material living or dead is present with proper warmth and moisture.Fungi
The plant body is unicellular or multicellular known as thallus which is not differentiated into root, stem and leaves. The thallus is simple, primitive, unicellular or multicellular and is known as mycelium.
Algae possess the green colouring matter known as chlorophyll. Fungi never possess chlorophyll.
Algae are autotrophs i e., they can prepare their own food from simple inorganic materials with the help of chlorophyll and photosynthesis. Fungi are heterotrophs i e., hey can not prepare their own food. They secure their food either directly or indirectly from other organisms. They as such live as saprophytes or parasites.
They cannot live in darkness, as light is absolutely necessary for the preparation of their food. They can live in light as well as in darkness.
The cell wall in algae are mainly composed of carbohydrate cellulose. The cell wall in most of the fungi are composed of chitin.
The food reserve in algae is frequently in the form of starch The reserve food is accumulated in the form of glycogen and oil globules.  
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