I want to discuss why we read Shir Hashirim on Pesach, and then to talk about a halacha which relates to that discussion.
There are two reasons we read Megillat Ruth on Shavuot. The thematic reason is that Ruth's acceptance of the Torah parallels our own acceptance at Sinai. The contextual reason is that the main events in the book took place during the harvest season ("She stuck with Boaz' maidens in order to glean until the end of barley harvest and wheat harvest", 2:23), which is around Shavout.
Regarding Shir Hashirim, the thematic reason for reading it on Pesach is clear. Pesach was the beginning of the love affair between God and the Jewish people. See for example Yirmiyahu 2:2 which builds on this theme ("Thus says Hashem: I remember for you the affection of your youth, the love of your marrying; how you went after Me in the desert, in a land that was not sown.").
Less obviously, there may be a contextual as well as a thematic reason for the Pesach reading. As with Ruth, one verse of Shir Hashirim alludes to a specific time of year. This is verse 6:11: "I went down into the nut garden, to look at the green plants of the valley, to see whether the grape vine had budded, and the pomegranates were in flower."
The phenomenon of trees, including the grape vine and pomegranate, flowering in spring is well-known. There is even a special blessing for it: "Blessed are you Hashem king of the world, who did not leave his world lacking in anything, and created in it good creations and good trees so that people may enjoy/benefit from them." (Shulchan Aruch OH 226). This blessing is recited when seeing flowering trees in the month of Nisan. The time of flowering may vary from species to species and place to place, but apparently, in Israel most tree species flower in Nisan. Thus the verse from Shir Hashirim is probably set in Nisan, and therefore, we read Shir Hashirim in the middle of Nisan - on Pesach.
Why does this blessing on flowering trees exist? Have you ever seen flowers that were so beautiful you could not avoid remarking on their beauty? It seems that the aforementioned blessing is an example of such a remark, only made to God (who made the flowers) rather than to a person. Yet while you might enjoy the beauty of any flower, for example tulips or daffodils, the blessing only applies to flowering trees which create edible fruit (see Be'er Hetev 226:1). It seems that to qualify for the blessing, the flowers in question must be not only beautiful but useful.
Perhaps this qualification comes from the difference between modern and ancient society. Nowadays, most of us live in cities, work in offices, walk on streets, and drive cars. We rarely see the plants, animals and landscapes of natural world like our ancestors did. This is not good for us and we try actively to fill the deficiency. We design parks, water plants and lawns, and take vacations for this reason. Perhaps for the same reason we are excessively fond of beautiful flowers, which in addition to their inherent attractiveness, bear the burden of representing nature in an environment where it is lacking.
For ancient people without advanced technology, things would have been somewhat different. Rather than having to seek out nature, they took it for granted as their all-encompassing surroundings. Like a long-married couple and unlike lovers who have recently met, they were familiar with nature and comfortable with each of its distinctive qualities. Flowers were one such quality, and to enjoy them was to appreciate one especially nice facet of the world they lived in, not to grab hold of a distant reality which they had long been deprived of.
Alongside this calmer approach to natural beauty came a heightened dependence on natural produce as the basis of one's livelihood. Thus flowers which looked nice were worthy of some interest, but flowers which looked nice and gave rise to grapes and pomegranates were of extreme interest. Neither vital but ugly sheaves of grain, nor pretty but useless violets, merited a blessing. But when the qualities of beauty and bounty were combined, a threshold was passed and our sages decreed the recitation of a blessing. One reason alone was not sufficient for the recital of Ruth or Shir Hashirim, nor for a blessing on flowers. Only the confluence of two separate reasons was a compelling enough basis for the creation of a new ritual.