I first came across top-down and bottom-up learning in my high school psychology class. I didn't pay much attention to it since I didn't see many applications. However, recently I picked up a book called Black Box Thinking: The Surprising Truth About Success, which made me think of the two principles and how useful especially the bottom-up approach can be.
Unilever is a big company that produces toothpastes, shampoos, even food and so on. Look on the label of yours and it's a fair chance that you'll find their logo. They were manufacturing washing powder in their factory in Liverpool. For its production, they were using nozzles, which allowed hot chemicals to disperse into vapour and powder. But there was a problem in their design. They didn't work smoothly, kept clogging up and were inefficient. So they hired a team of high-profile mathematics professors all specialised in fluid dynamics and high-presure systems. Their job was to find the best possible nozzle design for optimal performance and production of the shower powder. After building complex statistical and mathematical models, they finally achieved a new design...which had the same problems as the original one.
This is an example of top-down learning and thinking. They were applying their knowledge to an area they thought they knew and came up with a design that would, in theory, perform better. But theory and practice are two different things.
Unilever then went in the opposite direction. They didn't hire another group of highly specialised people for this task, like physicists maybe. Instead, they hired probably the most unexpected group of people for the given task. Biologists. Their design process was interesting. They took the original nozzle and made 10 different versions of it. Each one with only slightly different properties. They tested them out and picked the one that performed best. Next, they made another 10 variations based on the best nozzle from the previous series. They picked the best one and repeated the process about 45 times.
After building 449 nozzles before they finally designed the most efficient one. We call this bottom-up learning. As biologists, they didn't know much about fluid dynamics and whatnot. Instead, they relied on the most basic principle of nature that they knew very well; evolution. In the process, after designing so many different nozzles, they learned nozzle design to the last detail. They started with nothing and beat the specialised mathematicians.
This story illustrates the two principles of learning in a great way. Before specifically defining both ways of acquiring knowledge, there is another important distinction to keep in mind - implicit and explicit knowledge. Implicit knowledge is the knowledge that one gains through incidental activities or without awareness that learning is occurring. It cannot be easily transmitted to others nor defined. It just is there.  Explicit knowledge is the knowledge that can be readily articulated, codified, stored and accessed. It can be easily transmitted to others. That's the knowledge most known to us from textbooks, books and so on. 
Top-down learning (also called explicit-to-implicit) is acquiring explicit knowledge first and then learning implicit knowledge on that basis. 
Bottom-up learning (also called implicit-to-explicit) is acquiring implicit knowledge first and then learning explicit knowledge on that basis. 
Each pool of knowledge that's known to us today is only the tip of an iceberg; that's the explicit knowledge that we can learn about. But what's not seen is the amount of implicit information not immediately explainable. Only after a certain number of observations and tests, was it possible to produce a concise and understandable amount of knowledge also known as explicit knowledge.
Now, let's return to the nozzle design example at Unilever. The mathematicians had an incredible amount of explicit knowledge about fluid dynamics. They all had PhDs, were professors and successful in their field. However, what Unilever needed wasn't a complex mathematical model of the nozzle design, but rather a design that would work. They didn't need the top-down approach, but the bottom-up one. It's the same in evolution. Nature and organisms don't need to be complex to work. People are not even the most complex organisms on planet Earth. What nature and organisms need is an efficient way of surviving and developing, preferably in the simplest way possible. That way, fewer things can go wrong.
That's exactly what the biologists at Unilever achieved. It was a nozzle design that worked. They needed time, patience and imagination for tweaking the design, but it was well worth it. And in the end, they reached a design that was the most efficient they could make. Maybe they weren't able to explain it in that immediate moment, but I'm sure there was an objective, explicit, explanation behind it.
But which is better? Is any of the two better? In theory, bottom-up learning should be better. Why? Because we are subjected to learning by ourselves and learning from experience. Unilever's biologists failed 449 times before designing a successful nozzle. Just as evolution required 4 billion years for the development of life as we know today. This way, we can understand a problem in greater depth and apply it in day-to-day life. It, howeer, takes more time than top-down learning, where the information is already given to you to use it.
In January 2019, when I first got the idea of starting a website, I first learned HTML (HyperText Markup Language - editing text) and CSS (Cascading Style Sheets - changing the appearance of the website), the two basic programming languages for developing websites. It was not easy, even though this was not the first time I was exposed to the two of them. Yes, I worked through a book about the two, but I learned the most by trying to actively develop a website. That meant writing my HTML and CSS from scratch (without templates), including developing it for smartphones (implicitly). It was an incredibly time-consuming task. But that's how I learned CSS into detail and am now able to design a website from scratch.
This was now the time to use templates (or frameworks) that allowed me to develop the website in one-third of the time and was even better than when I have done it myself (explicitly, in a way). But, this wouldn't be possible without first knowing its core principles. Only then was I able to use it to the full extent.
This is an example of a bottom-up approach I've used (and am still using) to learn an area from its basic principles. It's not an approach to use in every area. Imagine studying anatomy or physiology from scratch, without textbooks and research papers. Or without basic knowledge of chemistry, physics and biology. Any complex subject is built from its component parts that we as students acquire throughout the course of our education.