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Comparison between convergent thinking and divergent thinking
The two sides of a coin
Both convergent and divergent thinking play important roles in finding the best solution to a problem. Convergent thinking is often used in accordance with divergent thinking.
The terms convergent thinking and divergent thinking were introduced by a psychologist J.P. Guilford, in 1967. The distinction between these two types of thinking was covered in his psychometric study of human intelligence.

According to him, there are two ways to find a solution. The straight-forward way, where you find a correct solution by thinking straight, to-the-point, having complete focus on the problem, known as convergent thinking, and the indirect way, in which you analyze the problem, think of all the possible ways to reach a solution, and generate creative ideas to solve it, known as divergent thinking.
Convergent thinking
Convergent thinking
In simple words, convergent thinking is the ability to find the correct solution, without having a creative approach. A person is a convergent thinker when his sole aim is to find the best correct answer to a question. He will think in a way that is direct, precise and time-effective. He will use his knowledge on that topic, or refer to other solved answers and improvise it to get the best suitable solution, or create one of his own depending on the stated facts and information on the given problem. Convergent thinking comes to people who know to set their consciousness, focus and targets on the problem they are solving.

Convergent thinking is highly preferable when methods are required to be reapplied and worked on again in a better manner to get a single unambiguous answer. The most important thing with this kind of thinking is that, the solutions come very quickly and precisely.
Divergent thinking
Divergent thinking
The ability to think out of the box is divergent thinking. A divergent thinker thinks in a way that would include all the possibilities to get to a solution, giving creative ideas to make it work. This kind of thinking comes in a free-flowing manner, with no limitations to time and imaginations. The bondage of time doesn't appear here, giving the person complete freedom to think about innovative and extraordinary ideas producing new results every time. You may or may not get the best solution, but various other ideas are sure to be conceived.

Divergent thinking comes naturally to a person who has a creative mind, but can also be induced and put into practice by the art of relaxing one's mind with sleep, humor or even alcohol! Divergent thinking is related to creativity, and is always required in creative fields.
Convergent Thinking Vs. Divergent Thinking
"The uncreative mind can spot wrong answers, but it takes a very creative mind to spot wrong questions." - Antony Jay
Convergent thinking is used in accordance with divergent thinking. When new, creative, innovative, and vague ideas are presented as a result of divergent thinking; systematic, procedural, precise and time-effective convergent thinking is required to work on the results of divergent thinkers, scrutinize and reach an unambiguous answer.
One example of convergent thinking is school. It requires us to learn things in order to get good scores in exams. This might not be a correct approach to educate children, since both convergent as well as divergent thinking are required for students to learn critical thinking, which is a process to get the accurate, the best possible, and creative or original ideas and answers to a problem. Critical thinking helps solve not only difficult work-related problems but also day-to-day life-related issues. It is important to not omit the various possibilities while trying to solve a problem, and equally important is to have acquired knowledge, that comes from learning things in school.
There are several ways to test convergent thinking, IQ tests being one example. Understanding the level of knowledge one has is possible, but that of creativity is not. There is no test that can grade your level of creativity, and hence, that of divergent thinking. It requires thought experiments to be conducted within one's mind. Thought experiments are mind mappings of different scenarios to think and recollect the possibilities created.
Examples
1. Convergent thinking -
Which one of the following words rhymes with 'boat'?
goat
phone
stroke
shake

Explanation: Here, the answer is given among a few wrong words. This does not require much thinking, but the knowledge to know the pronunciation of the word 'boat', and which other word would have a similar pronunciation among the given options. The wrong data can be easily ruled out if there are several options given, and if you have the required knowledge.

Divergent thinking -
Can you list the words that rhyme with 'boat'?

Explanation: Here, the answer needs to be found out. This will require thinking, trying out different words that would have the same rhyming pattern as 'boat'. This would require time, a lot of thinking, and would certainly give many answers.

Divergent thinking is creatively creating new possibilities to a solution. Like in this case a person will think of the possible rhyming words.

Convergent thinking is fetching data and facts from the knowledge you can acquire from any resources and apply them on the new possibilities you just created. Like for the possible words you thought of in your mind, you apply your knowledge and choose the most appropriate ones to rhyme with boat.

2. Convergent thinking -

My phone is giving me problems. I should probably get a new one.

Here, I am ruling out the possibility of my phone getting repaired and functioning properly.

Divergent thinking -

My phone is giving me problems. What can possibly go wrong? I'll get it checked, if it still doesn't work, I'll maybe buy a new one.
Here, I am thinking of the possibility of repairing my phone, before making a decision to get a new one.