Realistic quantum devices are inherently open and often involve environments with memory. Here, we investigate quantum resource generation in two classes of random circuits, namely, memoryless open and memoryful open random circuits, and compare their behavior with the well-explored random unitary circuit model. We show that environmental memory qualitatively alters the dynamics: while unitary and memoryful circuits exhibit sustained growth and saturation of entanglement and non-stabilizerness (magic); memoryless dynamics leads to a distinct behavior where entanglement decays to zero after transient growth, even though non-stabilizerness remains non-zero, indicating the persistence of nonclassical features beyond entanglement. Interestingly, within the system size and timescale considered in our work, we observe that memoryful circuits more effectively approach low-order quantum-state k-designs than the other two circuits. Finally, we show that the Krylov complexity reveals suppressed spreading of quantum states in memoryless circuits, in contrast to strong growth in unitary and memoryful dynamics, which saturates at the maximum value. Closed dynamics are therefore usually the most resource-generating, but are ideal; realistic dynamics are open and seem to generate less, but if they possess memory, they can sometimes even outdo closed dynamics.