Gottsched was in some ways a more commanding figure, especially physically, which may have something to do with why he is more likely to be heard of today than is Bodmer. Lessing reacted pretty harshly to Gottsched's neoclassic literary recommendations in 1759, after which it was felt that Gottsched's time as arbiter of taste was over. Later, Goethe, in his autobiography, penned a satiric portrait of Gottsched, and this view has pretty much stuck.
Bodmer must have had a milder character, and seems to have been much loved by his friends and "pupils," including the painter Heinrich Füßli, or Henry Fuseli, as he was known after he emigrated to England. The portrait of Fusseli with his mentor indicates the reverence with which Bodmer was held by those who knew him. Among his literary acquaintance and correspondents were the poets Wieland and Salomon Gessner; the critic and philosopher Johann Sulzer; Johann Casper Lavater, author of the famous "physiognomic fragments" (to which Goethe contributed); and the educational reformer Johann Heinrich Pestalozzi.
Bodmer was instrumental in launching the career of Friedrich Klopstock, but there is much more to his literary influence. Important in connection with my current research, he is probably the principal conduit for the transmission of the aesthetic concept of the sublime into German letters. Later, he translated John Milton's epic Paradise Lost, and his enthusiasm for Milton was the bone of contention between him and Gottsched, who found the Englishman's flowery language too "Baroque." If Gottsched represented the influence of French poetics in German letters, it was Bodmer who introduced English literary currents. Being from a "republican" nation, he felt great admiration for England. Peter Hanns Reill, in his book on the German Enlightenment and the rise of historicism, has noted that the early journal published by Bodmer and Breitinger, The Discourses of the Painters, was much influenced by Joseph Addison's Spectator essays, especially its "cosmopolitan description" of English customs.
When the Sturm und Drang era of German letters began, in the late 1760s, Bodmer was pretty much regarded by the young shakers and movers as superannuated, but this literary movement was precisely a reaction against the French neoclassical poetics recommended by Gottsched. Sturm und Drang writers, including Goethe, had by now discovered the liberating effect of English writers, including Shakespeare and Ossian, a discovery that owes much to Bodmer's pioneering efforts. This image of Bodmer as a young man shows, however, the different time in which he and Goethe came of age. This may, however, have been the way Bodmer looked when Henry Fusseli first knew him.