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The most likely diagnosis for this patient is intermittent claudication. This condition is characterized by pain or cramping in the legs that occurs during physical activity, such as walking, due to inadequate blood flow to the muscles. The fact that the pain improves with rest indicates that the muscle demand for oxygen decreases when the patient stops walking, allowing for temporary relief.

In individuals who smoke, the risk of vascular problems increases significantly, contributing to peripheral artery disease, which can lead to intermittent claudication. While both peripheral artery disease and intermittent claudication are related, the latter specifically describes the symptoms experienced by the patient during activity and their resolution with rest.

Other conditions, such as deep vein thrombosis, would likely present differently, often with swelling and pain at rest, while Raynaud phenomenon involves episodes of cold-induced vasospasm, which is not consistent with the symptoms described. Thus, the context and symptom pattern strongly support the diagnosis of intermittent claudication due to its characteristic features of exertional leg pain relieved by rest.