According to MacTutor, "the subject was known as analysis situs for many years and only in the late 1920s was the English word topology used by Lefschetz". Lefschetz's 1924 work is titled Analysis Situs and Algebraic Geometry, but his influential 1930 monograph is already Topology, which he defines as "doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity..".
"Why" is speculative. From an answer on MSE:"Lefshetz [Topology, Amer. Math. Soc. Colloq. Publ 12 (1930), page 361] wrote that Poincaré had tried to develop the subject along `analytic' lines, but had turned instead to combinatorial methods because the analytic approach failed for example in the Poincaré duality theorem". Since the focus shifted from analytic to combinatorial methods "analysis" was not a fitting name. Listing published a book Vorstudien zur Topologie in 1847 and used the word since 1836. According to his 1883 obituary it was meant to distinguish "...qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated", which clearly resonates with Lefschetz's description. So it was a natural choice to make a point. Later, a similar change in perspective was reflected in the switch from "combinatorial topology" to "algebraic topology", attributed to Hopf, under Noether's influence, Mayer and Vietoris.