Consolidation stabilizes a new memory; reconsolidation restabilizes an old one
Consolidation is the process by which a freshly acquired memory becomes stable and durable. In the mammalian brain, new memories initially depend on the hippocampus for maintenance and recall, and over time the neocortex takes over, making the memory hippocampus-independent; this transformation is called systems memory consolidation [5]. Reconsolidation is different: reactivating a well-consolidated memory can temporarily return it to a hippocampus-dependent state, a phenomenon called systems reconsolidation [5]. So consolidation concerns a memory that is still new, while reconsolidation concerns an established memory that has been destabilized by reactivation and must be stabilized again.
This distinction matters for the sleep question because the two processes occur at different points in a memory's life. Consolidation happens after initial learning; reconsolidation happens after a reminder or retrieval that makes an old memory labile again. Whether sleep matters for each therefore depends on when the sleep opportunity falls relative to learning or reactivation.
Sleep is a well-established supporter of consolidation
The idea that brain activity during sleep is critical to consolidating newly acquired memories is a long-standing and central finding in the field [3]. Reviews of the topic describe consolidation and reconsolidation as components of a complex system that maintains the connectivity of memories across their lifespan, with sleep implicated in many of these processes [2]. The evidence base for sleep's role in consolidation is therefore broad and mature.
A computational model of systems consolidation and reconsolidation proposes that both processes can be accounted for by well-documented mechanisms of synaptic plasticity and stability, and derives predictions for testing them [5]. That model addresses the neural mechanisms rather than sleep specifically, so it does not by itself establish a sleep dependence; it is useful mainly for showing that the two processes are mechanistically distinct and can be modeled separately.
Reconsolidation also appears to involve sleep, but the evidence is newer and less complete
Direct evidence that sleep supports reconsolidation comes from starlings: sleep-dependent consolidation and reconsolidation both occur in this species, and sleep appears critical not only for initial consolidation but also for reconsolidation after a memory is destabilized [3]. This is a striking result because it extends the sleep dependence beyond new learning to the restabilization of reactivated memories, at least in birds.
In mammals, the picture is less settled. One review notes that sleep allows new memory traces to stabilize long-term and also seems to influence memory reconsolidation, while explicitly stating that much less is known about sleep's role in reconsolidation than about its role in consolidation [4]. The word "seems" is doing real work here: the influence is suggested, not firmly established. A separate study on motor memory found that pleasant and unpleasant emotion induced after retrieval, together with different exercise intervals, affected motor memory consolidation and reconsolidation [1], but its abstract does not isolate sleep as the factor, so it does not directly answer how much reconsolidation depends on sleep.
The practical difference: timing and strength of evidence
The clearest difference is one of confidence and timing. For consolidation, sleep is treated as a critical contributor supported by a large body of work [2][3]. For reconsolidation, sleep is implicated—most directly by the starling experiments [3] and suggestively by reviews of mammalian work [4]—but the evidence is described as limited, and the mechanisms are not well understood [4][5]. If you are asking whether a night of sleep after learning helps lock in a new memory, the answer is well supported. If you are asking whether sleep after reactivating an old memory helps restabilize it, the answer is a qualified yes: it appears to, at least in some species and paradigms, but the research base is much thinner.
One caveat sits beside the reconsolidation claim: the strongest direct demonstration comes from starlings [3], and the mammalian review language is cautious [4]. Whether sleep's role in reconsolidation is as general and as strong as its role in consolidation remains an open question rather than a settled finding.
Sources used in this answer
The effect of type and time of emotional induction and night sleep on consolidation and reconsolidation mechanisms of motor memory
Pleasant and unpleasant emotion induced after retrieval, and different exercise intervals, affected motor memory consolidation and reconsolidation.
Memory consolidation and reconsolidation: what is the role of sleep?
Memory consolidation and reconsolidation can be viewed as components of a complex system maintaining memory connectivity across the lifespan, with sleep implicated in many of these processes.
Sleep-dependent reconsolidation after memory destabilization in starlings
In starlings, sleep is critical both for consolidation of newly acquired memories and for reconsolidation after memory destabilization.
Reconsolidation of appetitive memory and sleep: functional connectomics and plasticity
Sleep allows new memory traces to stabilize long-term and also seems to influence memory reconsolidation, though much less is known about the latter.
A Computational Model of Systems Memory Consolidation and Reconsolidation
A neural model based on synaptic plasticity and stability can account for findings in systems consolidation and reconsolidation and yields testable predictions.