To "object" to something is "to feel or express opposition to or dislike of something or someone". When you "reject" something you "refuse to accept, use, or believe something or someone" (definitions from Cambridge dictionary) I'll give you a few examples. "The company rejected my job application." They refused to accept my job application. "They objected to my lack of experience." They disliked the fact that I didn't have enough experience. I hope this makes things clear for you.