Structural Grammar: A grammar intended to explain the working of language in terms of the functions of its components and their relationships to each other without reference to meaning. (educationengland.org.uk)

Structural grammar excludes semantics (construction of meaning) while analyzing individual phonological units of sounds (phonemes), the construction of words (morphemes and inflections), and syntax (function and relationship between sentence parts).

Structural grammar grows from Saussure's work in langue versus parole and diachronic language versus synchronic language. Therefore structural grammar analyzes concrete synchronic expressions of parole in order to find the abstract diachronic universalities of langue. That is to say, it analyzes individuals' concrete expressions of language at a fixed time (not over extended time) in order to understand the abstract (general, societally manifested) fundamentals of language that are constant over time.

Generative Grammar: A grammar intended to explain the universal qualities of language in terms of rules that underlie all grammatically possible sentences in any language and in all language with a distinction made between universal deep structure and individual surface structure.

Generative grammar began by exclusively analyzing syntax (function and relationship between sentence parts) though the field expanded to include analysis of phonology and semantics (phonemes and the construction of meaning). There is some overlap of generative grammar with structural grammar and some divergence.  

Generative grammar grows from the work of Noam Chomsky and is founded upon the proposition that a deep cognitive structure generates a syntactic base of all language and it consists of rules governing phrase structure that are implemented in transforming elemental rules into complex expressions. That is to say, there are naturally occurring cognitive rules for phrase formation that are naturally applied in the construction of complex expressions of sentences as language expressions. The transformations are expressed on the surface social structure as actual language. Various languages have the same deep structure, it is postulated, while each has a different surface structure expression. (Columbia Electronic Encyclopedia)

[The Columbia Electronic Encyclopedia, 6th ed. Columbia University Press, 2011.]