Active matter deals with systems whose particles consume energy at the individual level in order to move. To unravel features such as the emergence of collective structures several models have been suggested, such as the on-lattice model of run-and-tumble particles implemented via the Persistent Exclusion Process (PEP). In our work, we study a one dimensional system of run-and-tumble repulsive or attractive particles, both on and off lattice. Additionally, we implement a cluster motility dynamics in the on-lattice case (since in the off-lattice case cluster motility arises from the individual particle dynamics). While we observe important differences between discrete and continuous dynamics, few common features are of particular importance. Increasing particle density drives aggregation across all different systems explored. For non-attractive particles, the effects of particle activity on aggregation are largely independent of the details of the dynamics. On the contrary, once attractive interactions are introduced, the steady state, which is completely determined by the interplay between these and the particles' activity, becomes highly dependent on the details of the dynamics.