It seems to me that generative linguistics presupposes the structuralist tradition. Are there any major differences between these two methods?
The fundamental difference is that generative grammar purports to be a model of mental processes and (quasi-classical, non-Sapirian) structuralist linguistics denies that or is agnostic. Technically, GG is a perfectly explicit description of the competence of the ideal speaker-hearer (Aspects p. 4), but then there isn't much GG around, given the "perfectly explicit" desideratum. Also, bear in mind that many people equate GG with "Chomsky's current theory of syntax", which is a misunderstanding of the concept.
You cannot really compare structuralist and generative linguistics directly. In broadest terms, generative linguistics is one way to study and model language structure. It is therefore a part of the broader structuralist program (see for instance F Newmeyer).
In particular, generativism shares these structuralist commitments:
- Language as an interconnected hierarchical semiotic system
- Distinction between competence and performance (languge/parole)
- Meaning and function expressed through distinctive features
- Language susceptible to syntagmatic and paradigmatic analysis
- Specific concepts such as phoneme, morpheme, lexeme
- Language units as symbolic pairings of meaning and form
- Linearity of language
While many of the generativist approaches grew out of a critique of the structuralist orthodoxy of the time, they still stayed firmly within the broader structuralist framework. In some way, generative linguistics could be seen as a contribution to a structuralist theory of syntax.