supply 的常见表达为 ~ sb/sth (with sth) ~ sth (to sb/sth) 尤指大量的供给: Foreign governments supplied arms to the rebels. 外国政府向叛军供应军火。
provide的常见表达为 ~ sb (with sth) ~ sth (for sb) 意思是 “给予”“让别人能够使用”:The hospital has a commitment to provide the best possible medical care. 医院有义务和责任尽可能地提供最好的医疗服务。
通常情况下,只要你注意这2个词的搭配,就行了,属于近义词,可互换
参考资料:牛津高阶英英词典