Bismarck was tremendously disappointed when Napoleon III was captured by Prussian troops in the aftermath of the battle of Sedan: the capture of the emperor put the Bonapartist regime in jeopardy and removed the only party with whom he might have negotiated a quick armistice and peace treaty.
Bismarck's worries were quite real: the regime went down like a castle of cards, to be replaced by a provisional republican government (anathema for the old junker) and the war which was to be over in September 1870 dragged on until March 1871, with a peace treaty signed only in May.
It was quite a problem for Otto, since the mobilization was to last much longer than anticipated, with its associated expenses as well as the negative impact on German economy. More importantly, the longer the war lasted the more unsettled would the political situation become in Prussia and the rest of Germany and the more skittish the diplomatic situation in Europe.
Let's assume that the Franco-Prussian war goes on as per OTL up to the battle of Gravelotte, after which Napoleon still decides to send MacMahon to try and relieve Bazaine in Metz, The only change is that ITTL Napoleon does not accompany MacMahon but decides to go back to Paris (for whatever reasons: the most logical explanations would be that he realizes the fragility of the regime - Eugenie is quite hysterical and count Palikao is positively hated by the majority of Parisiens - or maybe his gallstones make him feel worse, or whatever).
What do you think is going to happen? There should be major buttrflies in France, the Germanies and Italy too (Roman question).
Let me put down a couple of rules: after Sedan (which is obviously another major Prussian victory) Napoleon does not flee from Paris; if Napoleon is going to ask for an immediate armistice and a negotiates in good faith a peace treaty Bismarck will give him one much better that the OTL treaty of Frankfurt