Advances in technology provide an opportunity to enhance the accuracy of gait and balance assessment, improving the diagnosis and rehabilitation processes for people with acute or chronic health conditions. This study investigated the validity and reliability of a smartphone-based application to measure postural stability and spatiotemporal aspects of gait during four static balance and two gait tasks. Thirty healthy participants (aged 20–69 years) performed the following tasks: (1) standing on a firm surface with eyes opened, (2) standing on a firm surface with eyes closed, (3) standing on a compliant surface with eyes open, (4) standing on a compliant surface with eyes closed, (5) walking in a straight line, and (6) walking in a straight line while turning their head from side to side. During these tasks, the app quantified the participants’ postural stability and spatiotemporal gait parameters. The concurrent validity of the smartphone app with respect to a 3D motion capture system was evaluated using partial Pearson’s correlations (rp) and limits of the agreement (LoA%). The within-session test–retest reliability over three repeated measures was assessed with the intraclass correlation coefficient (ICC) and the standard error of measurement (SEM). One-way repeated measures analyses of variance (ANOVAs) were used to evaluate responsiveness to differences across tasks and repetitions. Periodicity index, step length, step time, and walking speed during the gait tasks and postural stability outcomes during the static tasks showed moderate-to-excellent validity (0.55 ≤ rp ≤ 0.98; 3% ≤ LoA% ≤ 12%) and reliability scores (0.52 ≤ ICC ≤ 0.92; 1% ≤ SEM% ≤ 6%) when the repetition effect was removed. Conversely, step variability and asymmetry parameters during both gait tasks generally showed poor validity and reliability except step length asymmetry, which showed moderate reliability (0.53 ≤ ICC ≤ 0.62) in both tasks when the repetition effect was removed. Postural stability and spatiotemporal gait parameters were found responsive (p < 0.05) to differences across tasks and test repetitions. Along with sound clinical judgement, the app can potentially be used in clinical practice to detect gait and balance impairments and track the effectiveness of rehabilitation programs. Further evaluation and refinement of the app in people with significant gait and balance deficits is needed.