Weather is important primarily because the Igbo are an agrarian society. As such they are immediately dependent on weather for their livelihood, and their religious beliefs are connected to the role of the elements, with the “rain-maker” conducting rituals at particular times. Achebe likens severe deviation from the expected conduct of the world to madness. In regard to the novel’s plot, a bad year of drought alternating with excessive rain plays an important role. Despite Okonkwo’s tremendous work ethic, just putting in more time cannot save the yams (Chapter 3).

Okonkwo, determined to act on a large scale, obtains 800 additional seed-yams from Nwakibie and 400 from his father. Initially, he sows his own plants. Unfortunately, this turns out to be

the worst year in living memory. Nothing happened at its proper time; it was either too early or too late. It seemed as if the world had gone mad.

For two months, the people endure a drought, watching their crops wither and parch in the field. Okonkwo had held back some plants, and now he sows those. Now, the rain comes, but too much.

But the year had gone mad. Rain fell as it had never fallen before. For days and nights together it poured down in violent torrents, and washed away the yam heaps.

The ground becomes so saturated that trees are uprooted, and the torrents create deep gorges. Things have progressed so far that human intervention will have no effect (Chapter 4).

[E]ven the village rain-maker no longer claimed to be able to intervene. He could not stop the rain now, just as he would not attempt to start it in the heart of the dry season, without serious danger to his own health.

In Things Fall Apart, Unoka, Okonkwo's father, loves the weather and the changing of the seasons because he is an artist, not a farmer.  Okonkwo, though, wrestles with the weather like he wrestles "The Cat" and fights his wives and the white men.

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Unoka loved the good fare and the good fellowship, and he loved this season of the year, when the rains had stopped and the sun rose every morning with dazzling beauty. And it was not too hot either, because the cold and dry harmattan wind was blowing down from the north.

The repeated imagery of the west wind, the harmattan, in the first half of the book foreshadows the coming of the Westerners in the second half of the book.  Okonkwo struggles to grow yams to make up for his father's debt and to provide for his family and tribe.  Yams are difficult to cultivate, and the weather of West Africa makes matters worse.  But, Okonkwo is a better farmer than he is a husband, father, and leader of men.  He finds a kind of balance in nature.  So says Enotes:

As a farmer, he readily accepts the unpredictability of the weather, thus affecting his crops, but he cannot accept the unpredictability of human nature, whether it is his son or the British colonial officials.  Rage erupts when he is crossed.

How significant is the role of natural phenomenon/ weather in relation to other element in causing Okonkwo’s downfall in Things Fall Apart?

I would have to answer that the weather is not as significant as other factors in Okonkwo's downfall. The coming of the British and the changes they have wrought upon Ibo culture, in addition to Okonkwo's own stubbornness and arrogance, have a far greater impact on his life than natural phenomena. In fact, the only time weather plays a role is during times of drought. These dry seasons only strengthen Okonkwo's resolve. Although they create hardships for him and his family, he remains more determined than ever to prove a success. Yet this does not lead to his eventual suicide. That is a result of events over many years, mostly beyond Okonkwo's control. He could not stop the missionaries from coming to Umuofia, and try as he might, he could not stall their influence on members of the village. Some might argue that he himself could have changed, but the resoluteness was ingrained in his personality. So, it led directly to his tragedy.

Although it is not directly related to Okonkwo's situation, weather, particularly rain, is essential to the livelihood of the Ibo people. Without rain, the yams cannot grow, & without yams, a man cannot feed his family. Therefore he is not a man. For Okonkwo, the times of drought reflect a sterility in his heart. He feels no love for Nwoye, his first son, & he has difficulty expressing any kind of feelings for any other members of his family. These are the true causes of his downfall.