意思不同、用法不同。compete with是和...竞争，后加竞争对象。compete against是反对某人，与...抗争，与...竞争，后加对抗对象。
compete的基本意思是“力图取胜”,指在比赛、辩论等有竞争活动中为征服或取胜而进行的努力,常暗示有奖赏刺激物。
compete是不及物动词,接against表示“与…对抗”,接for表示“夺取…”,接with表示“对…抗争”。
compete against(v.+prep.)
与…抗争 struggle against sb or sth with hope of success
The firm is too small to compete against large international companies.这个公司太小,不能与大跨国公司竞争。
compete for (v.+prep.)
比赛…夺取 try to win sth in competition with sb else
Firms from across the country may compete for the contract.全国各地的公司都可能为承揽这项合同而竞争。
compete with (v.+prep.)
与…抗争 struggle against sb or sth with hope of success
〔说明〕 compete with常不用于进行体。
We try to make the trains run on time, but we can't compete with weather.我们努力使火车准点运行,但我们抗争不过天气。