We present a comparative study of ultrafast photoconductivity in two different forms of one-dimensional (1D) quantum-confined graphene nanostructures: structurally well-defined semiconducting graphene nanoribbons (GNRs) fabricated by a "bottom-up" chemical synthesis approach, and semiconducting carbon nanotubes (CNTs) with a similar bandgap energy. Transient photoconductivities of both materials were measured using time-resolved terahertz spectroscopy, allowing for contact-free measurements of complex-valued photoconductivity spectra with sub-picosecond time-resolution. We show that, while the THz photoresponse seems very different for the two systems, a single model of free carriers experiencing backscattering when moving along the long axis of the CNTs or GNRs provides a quantitative description of both sets of results, revealing significantly longer carrier scattering times for CNTs (ca. 150 fs) than for GNRs (ca. 30 fs) and in turn higher carrier mobilities. This difference can be explained by differences in band structures and phonon scattering and the greater structural rigidity of CNTs as compared to GNRs, minimizing the influence of bending and/or torsional defects on the electron transport.