What is the difference between morphs  and allomorphs?

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In morphology, morphemes are the smallest, indivisible units of a word that have lexical and grammatical meaning. Morphs and allomorphs are the phonological manifestation of a morpheme. A morph (from the Greek word morphē, which means "form" or "shape") represents the formation of a morpheme, or rather its phonetic realization; an allomorph presents the way that morpheme might sound when pronounced in a specific language or its phonological realization. In the book The Oxford English Grammar by Sidney Greenbaum, the differentiation between morphs and allomorphs is described as follows:

When a morpheme is represented by a segment, that segment is a morph. If a morpheme can be represented by more than one morph, the morphs are allomorphs of the same morpheme: the prefixes in- (insane), il- (illegible), im- (impossible), ir- (irregular) are allomorphs of the same negative morpheme.

For example, the English plural consists of three different morphs, which makes it an allomorph because there are several alternatives:

  • /s/ - cats
  • /z/ - gloves
  • [əz] – wishes

As you can see, the "s" makes a different sound in each of the above examples.

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Morphs and allomorphs are both ways to describe the phonetic expressions (that is, the actual sounds produced) of the smallest meaningful parts of language in the study of morphology: morphemes.

A word like "unhappy" has two meaningful pieces: "un" and "happy," which together convey the meaning "not happy." Each of the meaningful parts is a morpheme, the phonetic expression of which is called a morph. Morphs can be further classified into "lexical" or "grammatical": lexical morphs are the meaningful roots, like "happy" or "man," and these are often free-standing words. Grammatical morphs modify the root in a meaningful way, but may not stand as a free word; in English, suffixes like [-able] or prefixes like [un-] are grammatical morphs. 

Allomorphs are phonetic variants of a morph. A good example of allomorphy is the plural suffix in English, which can have the allomorphs [-s], [-z], or [-ez] depending on the phonetic environment.

One way of looking at this is that any allomorph is simply what you call a morph that has another possible phonetic expression. 

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