Azotemia refers to increased urea (and creatinine) in blood, while uremia, which literally means "urine in blood", refers to azotemia plus symptoms and signs of renal failure. So simplistically, azotemia is a biochemical abnormality in an otherwise asymptomatic individual while uremia is more clinically advanced in individuals with clinical features of renal failure.

Thanks @mogamat-yaziedchothia

Uremia refers to the clinical syndrome developing in chronic kidney disease, in regards to fluid, electrolyte and metabolic abnormalities. This term comes from the increased urea levels in blood, although clinical symptoms cannot be attributed solely to urea. On the contrary, urea serum concentration represents a marker of low-molecular weight solutes filtration. The clinical consequences of CKD are mediated by a variety of uremic toxins, both water-soluble and protein-bound. Phospate, PTH, FGF-23, β2-microglobulin, indoxyl sulfate and p-cresyl sulfate are mentioned as some examples from the long catalogue of uremic toxins, which still remains under investigation.