Very good question. In this case, these are not different influences over time, but the words are indeed related.
We see that in other languages as well (does a bank teller tells stories or counts money?) but in Hebrew there are some deeper origins to the relationships. Also to להסתפר in this case... for more details, read the Tanakh. :)
Maybe a bit of grammar will help here. Specifically, that of "מושא ישיר" (direct object; read: "Moosa Yashir") vs. "מושא עקיף" (indirect object; read: "Moosa Akif").
אני סיפרתי לו סיפור = I told (to) him a story
The verb "told" has an indirect relation to the "him". They are linked via the linking "to" (ל...). In comparison, the object "a story" has more direct connection to the verb: It's simply "I told a story" with no linking words (or sometimes with the word "את", which is a special little thing by itself, so for that matter doesn't count as a linking word).
More examples will probably clarify it better.
אני אכלתי תפוח = I ate an apple.
אני אכלתי את התפוח = I ate the apple
"an apple" or "the apple" are direct objects here.
אני אכלתי מהצלחת = I ate from the plate
"the plate" is an indirect object, connected to "ate" via the word "from" (מ...).
So, to your question, when using "את" (or nothing at all) it's probably a direct object (unless it's an older Hebrew...), while with "ל..." it's an indirect object. Examples:
אני סיפרתי את הסיפור = I told the story (direct object)
אני סיפרתי סיפור = I told a story (direct object)
אני סיפרתי אותה = I cut her hair (I'm a barber, different Binyan altogether, different meaning to the verb)
הסַפָּר סִיפֵּר לִי סִיפּוּר = The barber told (to) me a story (note the "לי", now "me" is an indirect object, "a story" is a direct object).