They don't really mean "I will," but rather "I will have" and "I will be," respectively. J'aurai is the future of avoir - to have. So J'aurai sûrement un quart d'heure de retard literally means "I will surely have a quarter hour of lateness." Here, retard is a noun in French, so you "have" it. Je serai is the future of être - to be. Je serai sûrement en retard d'un quart d'heure means I will surely be late by a quarter hour. Here, en retard is an adverb, so you are late. The difference is probably easier to understand with different examples: J'aurai du temps libre demain = I will have some free time tomorrow. vs Je serai là à midi = I will be there at noon.
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