Poetry – Can Poems Follow a Rhyme Scheme but No Metre?

Literacy, pens, paper, the printing press.

A written culture has different restrictions than an oral culture dependant on ease of repetition from memory.

According to the University of Wisconsin Digital Collections Center:

Beowulf is the oldest narrative poem in the English language, embodying historical traditions that go back to actual events and personages in fifth- and sixth-century Scandinavia. During the long preliterate centuries when these traditions were transmitted in the form of oral poetry, they were combined with with a number of legendary and folktale elements (among these are Grendel and his mother, the dragon, and probably the hero Beowulf himself). The written text of the poem, as we have it today, took shape in England during the middle or late Anglo-Saxon period and survives in a single manuscript from around the year 1000.

An oral tradition requires stories to be easily memorised and stand repetition many, many times, and passed on to the next storyteller. A strong metre and fixed structure with helps, along with alliteration (also found in Beowulf) and isn't unique to English. For example, the Finnish national epic, the Kalevala, is based on sung oral tradition and has a fixed pattern of stresses, with much alliteration and parallelism (repeating the previous line with different words but meaning the same thing). In fact, the hero, "steady old Väinämöinen" (an almost always alliterative "vaka vanha Väinämöinen"), is himself a storytelling wizard who plays a zither and uses his song-words for magic.

Once people can read and write, they no longer have a need for the storyteller to recite a story from memory, they can read it themselves, or have someone read it to them from a text. Over time, this gives rise to more creative ways of expression.

For more, here's a paper (PDF) on Oral Tradition & Its Decline by Indira Bagchi.

The names come from French, where (from French Wikipédia)

A rhyme is called feminine when the last phoneme is a mute e (formerly called an "e féminin").

That is, a rhyme was called feminine if the words ended with a mute e. Back when the rules for French poetry were formulated, these e's were pronounced, but unstressed, and one name for them was feminine e's. Mute e's are still pronounced when reading poetry and when singing, although not in normal speech.

Why were they called feminine e's? It probably didn't have anything to do with mute e's being weaker or more girly-sounding in any sense; they were called feminine e's because, to turn male adjectives and some male nouns into female ones, you added a mute e. For example, in French, a big black cat is:

un gros chat noir (boy cat),
une grosse chatte noire (girl cat).

This rule in French applies only for adjectives and some nouns which have male/female versions (e.g., chanteur, chanteuse); there are quite a few masculine nouns that end in mute e's and feminine nouns that don't.

You can see that the name "feminine rhyme" originated in French by looking at the reference (from London, 1764) T Romano gives in his answer, where a feminine rhyme is defined as one ending in an e-mute. In English, words (e.g., state and gait) rhyme whether or not they end in a silent e.