TL;DR: I think the two 'teams' in question are...
...a rope and a chain. If you consider these items in terms of their basic components - usually natural or synthetic fibres, and metal bars bent into hoop shapes, respectively - the chain would appear to be the much stronger of the two. However, there is a famous proverb that "a chain is only as strong as its weakest link" (often used in the context of a team only being as strong as its weakest member). In other words, because a chain comprises a series of linked metal hoops, each connected one-to-the-next in a single line, it just takes one link to break for the whole chain to become useless. In contrast, ropes are made from several fibres twisted and braided together, providing it with tensile strength and meaning that if one fibre should snap the rope as a whole does not immediately fail.
In actual fact, a rope (especially a wire rope) may even be stronger than a chain, breaking the 'myth' (or misconception) that a chain must be stronger because the raw material from which it is made is stronger (metal vs fibres).
Stanza by stanza...
A proverb and a myth, I will tell you a tale of two teams.
I saw them last night in my dreams.
The one team was sleeker, but started much weaker,
The other was strong from the start.
The teams are a rope and a chain. The rope is 'sleeker' in appearance (not as bulky as the chain) but the properties of its raw materials are not as strong as (i.e. weaker than) the chain's when deconstructed - fibres vs metal ('strong from the start').
Yes, the other was strong from the start.
But we found that they had less heart.
They grew and grew by one link and two,
And then they came apart.
Chains are made of links, and have 'less heart' in that there are holes in the middle of each link, whereas a rope is one solid unit all the way along its length. You can make a chain longer by adding more and more links.
It was too late, broke under their weight.
But the other can't suffer this fate.
The sleeker was weaker, but it was a keeper,
They shared the load with each part.
In this parable, the chain breaks at a particular tension because as soon as the tension becomes too great for its weakest link, that link will break and the whole chain will become broken and useless. Meanwhile (as explained in the first spoiler) if a strand of a rope were to snap under tension the whole rope will not immediately fail, since each of its strands shares the load along the whole length of the rope.
They grew and grew, by one foot and two,
And bound their strength anew.
Silly as it seems, this tale of two teams,
A myth and a proverb, true.
You can make a rope longer (and capable of taking more strain) by splicing in additional threads.
Though it may seem grand, or from foreign land,
You’ve worked with them hand in hand.
Ropes and chains are humble everyday items that have been in use for centuries, in pretty much every country of the world. Doubtless you've probably used one yourself at some point, holding it in your hand while using it...
