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Galaxy morphology and its evolution over the cosmic epoch hold important clues for understanding the regulation of star formation (SF). However, studying the relationship between morphology and SF has been hindered by the availability of consistent data at different redshifts. Our sample, combining CANDELS (0.8 < z < 2.5) and the GALEX-SDSS-WISE Legacy Catalog (GSWLC; z ~ 0), has physical parameters derived using consistent SED fitting with flexible dust attenuation laws. We adopt visual classifications from Kartaltepe et al. 2015 and expand them to z ~ 0 using SDSS images matching the physical resolution of CANDELS rest-frame optical images and deep FUV GALEX images matching the physical resolution of the CANDELS rest-frame FUV images. Our main finding is that disks with SF clumps at z ~ 0 make a similar fraction (~15%) of star-forming galaxies as at z ~ 2. The clumpy disk contribution to the SF budget peaks at z ~ 1, rather than z ~ 2, suggesting that the principal epoch of disk assembly continues to lower redshifts. Star-forming spheroids ("blue nuggets"), though less centrally concentrated than quenched spheroids, contribute significantly (~15%) to the SF budget at z ~ 1-2, suggesting that compaction precedes quenching. Among green valley and quiescent galaxies, the pure spheroid fraction drops since z ~ 1, whereas spheroids with disks (S0-like) become dominant. Mergers at or nearing coalescence are enhanced in SFR relative to the main sequence at all redshifts by a factor of ~2, but contribute $\lesssim$5% to the SF budget, with their contribution remaining small above the main sequence.