What is the difference between amid and amidst?

Both words are close synonyms.

In day to day usage, amid is popularly used.

However, when it comes to literary usage, amid and amidst are both used by the British English speakers without a problem. However, the American English speaker prefers the term amid to amidst. It is because amidst with its extra –st sound at the end, sounds more as a word straight out of a Shakespeare's drama

Examples of their usage:

He could move slowly amid the crowd.

The minister reached the place amidst tight security.

Although both words are technically interchangeable based on individual preference, in my personal experience in the southeastern US, "amid" is used almost exclusively for positional reference (i.e. inside a crowd, between buildings, etc), while "amidst" is used for time-span or non-physical reference (i.e. during the confusion, over the course of his studies).

From www.etymology.com:

amidst (prep.) a variant of amid (q.v.) with adverbial genitive -s and unetymological -t. Amidde became amyddes (13c.) and acquired the -t from mid-15c., probably by association with superlatives in -st; the pattern also yielded amongst, against, betwixt, whilst, also archaic alongst (13c.-17c.).

There is a tendency to use amidst more distributively than amid, e.g. of things scattered about, or a thing moving, in the midst of others. [OED]