Both are general terms in botany. Since algae (sigular alga.)are more primitive than the mosses; I deal with them first and then I will Take up moss.
ALGAE-:
1)All of them are thallophytes as the plant body has no root , stem and the leaves.
2) All of them are aquatic; either fresh water or marine.
3) All of them are autotrophic as they manufacture their own organic food with the help of sun light and chlorophyll.
4) The process of sexual reproduction is either isogamous and anisogamous conjugation or oogamous fertilization.
5) On the basis of the dominant photosynthetic pigment, they are put in different divisions such as Chlorophyta (green algae), Phaeophyta(brown algae), Rhodophyta(red algae) etc.
MOSS (plural Mosses)-:
1) From the evolution point of view they are higher than the algae and most likely evolved from them.
2)Some of them show (especially mosses) root like, stem like and leaf like organs but these are not true roots, stem and leaves.(reasoning will require deeper knowledge of botany on your part).
3) All of them require moist habitat but will refuse to grow in water. There are no marine or salt water mosses or their relatives.
4) All of them are autotrophic in nutrition like algae.
5) The process of sexual reproduction is higher than that of algae and it is always oogamous fertilization(male and female gametes are very distinct from each other like those in animals).
6) All the mosses and the likes are put in a single division Bryophyta and all of them are referred to as the bryophytes.
Mosses and liverworts are the examples.
