A lot can be said about the difference between an incubator and a climatic chamber.
Both are at the center of research and experimentation and are widely used today in several applications.
For clarity’s sake, however, the differences may be grouped simply:

incubators create environments; climatic chambers simulate extremes.

From this basic difference, a vast array of things follow, which can summed up by saying that incubators are generally used to grow cultures and climatic chambers are for industrial testing.

Incubators create environments and are usually used to grow cultures

Rudimentary incubators have been profusely used in history.

At its core the idea is indeed simple and timeless: to create a specific environment for growth.

The tool has to keep certain conditions of temperature, CO2, oxygen, amongst others, towards doing experimental work.

The main purpose, therefore, is to create a stable environment, with controlled variables so that such trial may proceed.

There are several types of incubators, some quite simple and others more complex where more variables can be controlled, but the basic idea is the same:

(1) general purpose devices, may include forced convection or natural convection;

(2) carbon dioxide incubators, to create situations for natural cellular growth;

(3) refrigerated incubators, when there is a need for below ambient temperatures;

(4) shaking incubators, to provide agitation, ideal for liquid cultures for instance;

(5) BOD incubators, for specific biochemical oxygen needs.

Today, however, the market offers versions which combine the aforementioned functions into one product.
Incubators come in a variety of sizes, from tabletop devices to room-sized.
In general, they are not meant for use with hazardous substances.
Think of them as instruments for growth: tissue engineering, gene therapy, immunotherapy, stem cell research, etc.
All these so-called ‘advanced therapies medicinal products’ will use incubators in experimentation.
Agricultural research is, evidently, another field where incubators will be used.
The tool is, in essence, the heart of laboratory culture.

Climatic chambers simulate extremes and are usually used for industrial testing

A climatic chamber –also called an environmental chamber– is basically an enclosure to test the effects of particular conditions on biological items, industrial products, materials of different sorts, electronic devices, etc.

A chamber can have different sizes, depending on the types of testing required: some are large enough to walk into.
Different versions come with different functions; some may have digital outside screens.
They may also vary according to the needs of the particular industry using them.
A climatic chamber may be used for individual testing on particular items, preparation of specimens for further tests of a different sort or replication of environmental conditions of other tests.
The idea is not only to expose components to different circumstances but also to increase effects to understand their impact, testing extreme conditions.
And to do all this with safety, of course.
The difference between an incubator and a climatic chamber types of testing can be climatic chamber testing or thermal shock, for example.
The idea is to put the items inside the chamber to withstand the chosen environmental conditions.
A climatic test is designed for simulation of an environment, exposing a particular item to such conditions.
Quite similar to an incubator, for sure.
But thermal shock, for example, is aimed at simulating changes in extreme climatic conditions in short periods of time.
It is the possibility of simulating extremes which characterizes a climatic chamber.
There is a wide array of testing:

temperature and humidity
UV exposure
accelerated aging
water
salt/fog
vibration
shock
impact
wind/rain
flammability

amongst many others…
This testing is performed in a vast range of industries such as electronics, automotive, plastics, metal, toys, paper products, chemicals and many more.
Think of the climatic chamber as the industrial go-to to get to know products under all sorts of variables and experimentation.

Conclusions

To find out the optimal conditions to grow a particular crop, say Cayenne pepper, an incubator would be used to run an experiment.

The aim would be finding the particular environment most suited to this particular seed.
On the other hand, to find out how a camera would fare in a place like Siberia, with its extremely low temperatures or in Death Valley with its high degrees, a climatic chamber would be preferred.
A test could go as low as -94°F (depending if a single stage or double stage chamber is being used) or as high as 194°F, to test
how the product would do in extreme conditions.
Today’s market and consumers are much more demanding, which requires merchandises to be tested to the extreme; we never know: a product may very well end up exposed to severe environments so we have to be able to anticipate this possibility
As mentioned at the beginning, the difference between an incubator and a climatic chamber is: the incubators create environments; climatic chambers simulate extremes.
Incubators and climatic chambers seem similar on paper but climatic chambers could be described as incubators “on steroids”, with a more varied functionality and a focus on extreme experimentation to prepare products for the market.

Climatic chambers are fundamental for preparing the product to be tested in all temperature variations around the world.