Almost and nearly have similar meanings. They are both used to express ideas connected with progress, measurement or counting.

He slipped and almost fell to the ground. OR He slipped and nearly fell to the ground.
Dinner is almost / nearly ready. (= Dinner will soon be ready.)

Sometimes almost shows more ‘nearness’ than nearly.

Compare:

It is nearly 12 o’clock. (= perhaps 11.45)
It is almost 12 o’clock. (= perhaps 11.55)

Differences

Nearly can be used with very and pretty. Almost cannot be used with very and pretty.

I have pretty nearly finished. (NOT I have pretty almost finished.)

Nearly is less common in American English.

Differences

Nearly suggests progress towards a goal. Almost suggests ideas such as similarity.

He is almost like a father to me. (NOT He is nearly like a father to me.)
She has got a strange accent. She almost sounds foreign.

Nearly is not normally used before negative words like never, nobody, nothing, any etc. Instead we use almost or hardly with ever, anybody, anything etc.

The speaker said almost nothing.  OR The speaker said hardly anything. (= The speaker said only a few words of little importance.)

(NOT The speaker said nearly nothing.)

Almost nobody came. OR Hardly anybody came. (NOT Nearly nobody came.)