Perhaps more of a question for /r/science, but lets consider it to be part of history of science, shall we?

Darwin proposed that each generation produced variation, and that this natural variation resulted in individuals which were unequally fit to the local environment. The fittest bred more succesfully and brought more offspring to maturity. The least fit, less so.

Lamarck proposed that pressure from the environment brought forth the emergence of new characters in individuals adapted to these pressures, and that these newly acquired traits were transmissible to their offspring (His classic example beeing giraffes progressively stretching their necks by reaching ever higher leaves).

Notice that extinction plays no central role in either theory, but can accounted for by each. Each model allows for species to change through time and for different forms to appear and then be replaced by new ones at a later period. Also, neither one requires or accounts very well for mass extinctions as we know them today.

As to proof, although no experimental support was found for the transmission of epigenetic changes to offspring, the debate was decisively settled once the work of Gregor Mendel came to the fore. Mendel discovered the laws of inheritance and brought the concept of coded information, genes, dominance and recessivity into the debate. This brought the notion of variation out of the abstract by giving it a concrete vessel subject to objectively measurable laws. However, his work did not circulate much outside of Germany between the 1860s and the early 1900s, and when it was rediscovered independantly by DeVries, Correns and Tshermak somewhat coincidentally with early work on chromosomes and their significance. During those years, there were at times acrimonious debates between Darwinist and Lamarckians. Arguments on either side often would not meet the current criteria of valid scientific inquiry, and sometimes appealed to faith or even nationalism.