The quantum statistics of atoms is typically observed in the behavior of an ensemble via macroscopic observables. However, quantum statistics modifies the behavior of even two particles. Here, we demonstrate near-complete control over all the internal and external degrees of freedom of two laser-cooled Rb-87 atoms trapped in two optical tweezers. This controllability allows us to observe signatures of indistinguishability via two-particle interference. Our work establishes laser-cooled atoms in optical tweezers as a promising route to bottom-up engineering of scalable, low-entropy quantum systems.