Why were the 747SP's trailing-edge flaps so much different than those of the -100/-200?
The wing was relatively too big for the airplane after cutting bits of the fuselage off, which reduced weight by 11,000 lbs. The aircraft did not require the triple slotted high lift devices anymore: if the wing area is large enough the wing requires no flaps &amp; slats but has relatively high cruise drag. All in all the total weight savings on the SP was 45,000 lbs. (Source)

The B747SP could not do completely without flaps, but the simpler flaps of the SP saved weight. The competition at the time was DC10 and L1011, which had lower wing area and one less engine, resulting in better fuel economy. 


B747SP: 300,000 kg, wing area = 511 m$^2$, wing loading = 587 kg/m$^2$
L-1011: 231,000 kg, wing area = 320 m$^2$, wing loading = 722 kg/m$^2$
DC-10 : 263,100 kg, wing area = 368 m$^2$, wing loading = 715 kg/m$^2$
Source


In cruise, a smaller wing area means less resistance: higher wing loading gives better fuel economy, but creates problems at low speed which need to be solved with high-lift devices.