Please be careful of localized heavy rain.
この薬は局所的に塗ってください。
Apply this medicine topically.
部分的 also means 'partial', but 局所的 emphasizes being limited to a specific location or spot, whereas 部分的 can refer to any part of a whole.
局所的 means 'localized, confined to one spot', contrasting with 広範囲's sense of spreading over a large area.
local test methods
Formed from 局所 (local area) + 的 (adjectival suffix).