I think the difference between what is called Vedic religion and modern Hinduism is continuous
Agreed -- with the lack of a named founder, and with the considerable borrowing from ancient Vedic practices, the development of Hinduism had quite a bit of continuity.
I don't think this is accurate.
Of the
Upanishad, only 13 'Principal Upanishads' date from before the synthesis period, and even the oldest of those could not have been written before 800 BC. The other 200 or so 'minor Upanishads' all date from the synthesis period or later.
Of the epics, the
Mahabharta was compiled mostly after 400 BC; the
Bhagavad Gita dates from roughly 200 BC. The
Ramayana dates to 700 BC at the absolute earliest, and was mostly composed between 400 BC - 200 AD. Likewise, the Puranas all date from after 300 AD.
In other words, the primary elements of Hindu theology (not practice, but codified and
organized religion) date from the classical and medieval period. Genealogically, the ancient Vedic practices informed the rise of both Jainism and Buddhism, which flourished in ancient India for several centuries. Then the brahman (priestly) caste undercut Jain/Buddhist popular appeal by reforming Vedic religious practice to incorporate their insight and traditions. Both Buddhism and Jainism thus faded from India -- between the rise of Hinduism and the arrival of Islam, Buddhism all but died out in India until its revival in the 1890's (!) -- while Hinduism began to flourish.
I think treating Buddhism and Hinduism as practically prehistoric religions, or treating Buddhism as descended from Hinduism, does a disservice to both.