Both phrases are in similar definition, but indeed there is slightly difference. Both phrases mean always, since, having been, but 從來 emphasizes on the beginning, in the first place, while 始終 puts emphasis on the final result, in the end.
Example:
John從來不認識Mark。John has never talked to Mark before.
John始終沒機會認識Mark。John never has the chance to talk to John, even now.
However, 從來 and 始終 are usually interchangeable.
Also for the record here, there's a special category of phrases which is unique in Mandarin, we call it 偏義複詞 (compounds of partially semantic shift), compound words like these usually consist a pair of characters with opposite ideas and refer to only one idea out of the two, a semantic shift, so to speak. For example we would say 恩怨(favors & resentment), 好壞(good & bad), and 始終(beginning & end), and 恩怨 actually refers to resentment rather than favor, 好壞 means the bad rather the good, and 始終 emphasizes on the end rather than the beginning. 從來 isn't a semantic shifting compound word, it's just a compound word consisting an identical idea, 從 and 來 both mean the beginning here.