haggadah and aggadah

Pesach is coming up and we will be reading from the haggada הגדה. What is the connection between haggada and aggada אגדה - the stories found in rabbinic literature?

They both derive from the root הגיד - "he told, narrated", and so, according to Klein, can mean "telling, saying"  or "tale, narrative." Both aggada in general, and the haggada in particular are narratives that expound upon Biblical verses (although aggada has come to mean any non-halachic content in the Talmud and midrashim, regardless of whether or not they are based on a verse.) The haggada of Pesach has a particular connection to the verb, as it appears in the verse commanding the telling of the story of the Exodus -   וְהִגַּדְתָּ לְבִנְךָ בַּיּוֹם הַהוּא "And you shall tell your son on that day..." (Shemot 13:8)

But essentially, there was no real difference between hagada and agada, and you can find them used interchangeably. They both meant the same thing, and we find a number of words in Hebrew which are synonymous, but one starts with an alef and one with a heh:

הפליה אפליה - both meaning "discrimination"
החזקה אחזקה - "maintenance"
הזהרה אזהרה - "warning"
הונאה אונאה - "oppression, deception"

While both words are Hebrew, the words beginning with alef have more of an Aramaic influence.

As often happened in Hebrew, when we have two synonymous words, their meanings tend to diverge. So haggada came to be associated almost exclusively with Pesach. In Modern Hebrew, agada has also come to mean "folktale" or "fable", famously in the quote from Herzl (originally in German) - אם תרצו אין זו אגדה - "If you will it, it is no fable [aggada]." And aggadot are used to refer to stories for children. This was cause for opposition by some Haredi writers, who found this secular use showed disrespect for the aggadot of the Rabbis.

The verb הגיד higid comes from the root נגד. Klein writes that the ultimate meaning of this root is "to rise, be high, be conspicuous." So the verb higid, meaning "he made known, announced, declared, told", originally meant "he placed a matter high or made it conspicuous before somebody." This same root gives us the word neged נגד - "opposite", which again originally meant "that which is high or conspicuous." And the term nagid נגיד - "chief, leader, ruler", cognate with the Arabic najid, can also be understood in this light - "noble". Klein points out that the word nasi נשיא had a similar development  - literally "one lifted up" from נשא - "to lift."  Klein mentions an alternate theory by Barth that nagid originally meant "speaker, spokesman", and perhaps nasi also might have mean "speaker." In Modern Hebrew the title nagid is primarily used to for the governor of the Bank of Israel.