"Moss" gets blamed for a lot of green that is not moss at all. Telling them apart takes only a close look, and it matters because they want different things and respond differently to whatever you do about them.
True moss
A soft, leafy plant with tiny leaves arranged around a stem, forming cushions or carpets, very often with fine stalks carrying capsules held above the green. The leaves are usually a single cell thick, so they wet and dry in moments. If you can see little leafy shoots you could tease apart, and stalked capsules standing over them, it is a moss.
Liverwort
Either flat and ribbon-like, pressed close to the surface in branching green lobes (the thallose kind), or leafy and very low-growing with leaves in flat ranks. There are no upright leafy stems like a moss, and the spore capsules, when present, are quite different, often splitting into a little dark star. Liverworts favour really wet, often disturbed ground, such as the bare surface of pot compost, where they are a common sign of overwatering.
Lichen
Not a plant at all, but a partnership of a fungus and an alga living as one. Lichens are usually crusty, leafy or shrubby in texture and often grey, white, yellow or orange rather than fresh green. They feel dry and tough, not soft, and grow very slowly on bark, stone, gravestones and roof tiles. Reindeer "moss" is in fact a lichen, as is much of the grey-green crust on an old wall.
Algae
A formless green film or slime with no leaves, stems or structure at all, on damp paving, timber, glass and pots. If you can smear it with a finger and there is nothing leafy left behind, it is algae. It marks a surface that is simply too wet and too shaded.
Clubmosses
These look mossy but are larger, with tougher, scaly stems, and belong to an ancient vascular group quite separate from the true mosses. Mostly you meet them out on heath and hill rather than in the garden, and they are not the green you are likely to be puzzling over on a wall or lawn.
A quick field test
Faced with a patch of green and unsure what it is, a few seconds of looking usually settles it. Is it soft and leafy, made of tiny shoots you could tease apart, often with fine stalks standing above it? That is moss. Is it flat, lobed and pressed to the surface with no upright shoots? Liverwort. Is it crusty, leathery or shrubby, dry to the touch and a colour other than fresh green? Lichen. Is it a structureless film or slime you can wipe away in one smear? Algae. A hand lens turns these hunches into certainties, and once you have checked a dozen patches the answer comes at a glance.
Why the distinction is worth making
It is not mere pedantry. The four groups want different conditions and respond differently to whatever you do about them, so misidentifying the green is the first step to mistreating it. Algae signals a surface that is simply too wet and shaded and will wipe off; lichen is slow-growing and harmless and generally best left alone; liverwort on pot compost points to overwatering that wants easing; and moss, depending on where it is, may be a welcome carpet or a slip hazard to manage. Knowing which you have is what tells you whether to leave it, encourage it or clear it.