Disclaimer: This article was written for the German readership of the Terrarium Blog and thus, discusses the origin of the words ewiges Terrarium (meaning eternal terrarium) and Flaschengarten (meaning bottle garden). I've decided to translate this article nonetheless. If you're still interested, keep on reading.
Some call it bottle garden while others call it an eternal terrarium. But what's correct? There are actually no differences between the two terms. They both describe a hermetophere in a container, filled with plants and animals. Most of these containers are made out of glass, but aren't bottles. The term eternal terrarium is still quite young but describes the same as bottle garden.
Table of Contents
What is a hermetosphere?
The term hermetosphere comes from the term hermetic which originates from the latin term sigillum Hermetis. This translates to seal of Hermes. This seal was supposed to seal bottles in an airtight way. Consequently, hermetophere just means airtight sphere.
The Wardian case
The Wardian case was invented by the english medical doctor Nathaniel Ward in the 19th century to transport plants over long journey without harming them. In principle, it's a small, portable greenhouse. The Wardian case was especially interesting for botanists and plant researchers. With it, they could bring new shrubs and flowers on long journeys over the sea. Since it was hermetically closed, the plants were protected against dust and strong temperature changes and thus, could survive longer. It also made it possible to keep tropical plants in the colder and drier Europe.

Tropical plants became much beloved decorative objects in victorian England. But the Wardian case wasn't just that. It also had geopolitical implications. On the one hand, it was used to overthrow the Brazilian rubber monopoly by bringing rubber trees to Sri Lanka. Brazil and Sri Lanka have similar climate conditions but previous attempts to migrate the trees failed due to the harsh over-see journey without Wardian cases.
Read more about Wardian cases in this article.
Bottle garden vs. eternal terrarium
Like I said in the beginning of this article, the term bottle garden is much older than eternal terrarium. But they mean exactly the same. Since an eternal terrarium isn't always in a bottle, the term bottle garden probably isn't quite as accurate. If we compare the two on Google Trends, you'll see that bottle garden was googled way before eternal terrarium. Eternal terrarium has become a popular search term in April 2016. Many more people know the term bottle garden it seems, judging by search volume.

I also wanted to find out when the term bottle garden first got to the general public. For that, I used Google's Ngram Viewer. With it, you can see all uses of a word back to the year 1500. Sadly, I got disappointed. Apparently Google hasn't indexed any books with bottle gardens yet. Eternal terrarium also doesn't appear.
Why did eternal terrarium appear in April 2016?
Coining this term can be traced back to one person. In April 2016, the YouTuber SlivkiShow published a video about a closed terrarium in a light bulb. The video went viral and now has over 2.5 million views. It's probably still in many people's heads. Many people who've never seen the video will also use the term eternal terrarium today. Here's the video.
The search volume really only started after this video. Since then the term has a quite constant search volume. It's definitely possible that the term is here to stay. One day it might even overtake bottle garden.

Summary
Bottle garden and eternal terrarium mean exactly the same thing. They originated from the Wardian case that was invented almost 200 years ago. The term bottle garden has long been known. However, the term eternal terrarium just became relevant in April 2016 after YouTuber SlivkiShow coined it. Since then it has been in the general publics language. This is undisputably shown by Google Trends.
If you know want to know how a closed terrarium works, read this article here.