What does Lisp provide on top of e.g. the lambda calculus? I'd expect the latter to be even more fundamental. But I suppose one could reasonably argue that that wasn't 'Software', just Math.
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So to answer your question, nothing (in this context), because the Turing machine is a more useful model for thinking about computation itself.
Perhaps the same might apply to lisp / lambda calculus, although I suspect the answer here is that it's just a lot harder to right a self-interpreter in lambda calc.
Here's one for the http://en.wikipedia.org/wiki/Binary_lambda_calculus, all of 29 bytes long, including parsing:
0101000110100000000101011000000000011110000101111110011110
0001011100111100000011110000101101101110011111000011111000
0101111010011101001011001110000110110000101111100001111100
0011100110111101111100111101110110000110010001101000011010This isn't obvious to me. It seems to me that a TM is more useful for thinking about computation from an operational perspective. But this isn't the only, or arguably even the most effective, way to think about computation. The lambda calculus is much more useful for thinking about computation from a denotational perspective.