The answer as you say is "not much". If peers have some wealth associated with the peerage then I imagine it helps to differentiate yours as "old money" but I doubt many care much about that distinction. Whilst the peerage is not necessarily the cause, we shouldn't forget that hereditary peers are disroportionately influential in society, even if not as obvious as they once were.
For some it will matter to maintain family tradition, especially if there was a particularly famous forebear (usually the first peer, but not always).