Using itertools.dropwhile, itertools.takewhile, itertools.islice:

import itertools

with open('data.txt') as f, open('result.txt', 'w') as fout:
    it = itertools.dropwhile(lambda line: line.strip() != 'Start', f)
    it = itertools.islice(it, 1, None)
    it = itertools.takewhile(lambda line: line.strip() != 'End', it)
    fout.writelines(it)

Using itertools.dropwhile, itertools.takewhile, itertools.islice:

import itertools

with open('data.txt') as f, open('result.txt', 'w') as fout:
    it = itertools.dropwhile(lambda line: line.strip() != 'Start', f)
    it = itertools.islice(it, 1, None)
    it = itertools.takewhile(lambda line: line.strip() != 'End', it)
    fout.writelines(it)

UPDATE: As inspectorG4dget commented, above code copies over the first block. To copy multiple blocks, use following:

import itertools

with open('data.txt', 'r') as f, open('result.txt', 'w') as fout:
    while True:
        it = itertools.dropwhile(lambda line: line.strip() != 'Start', f)
        if next(it, None) is None: break
        fout.writelines(itertools.takewhile(lambda line: line.strip() != 'End', it))