It is surprising to see that some see mountaineering and mountain climbing as different activities. I also see that some describe them as sports. My views are quite different.
A direct answer to the question in the title is: mountaineering and mountain climbing are the same activities. Some people see these activities as a sport, and for mountain guides they are also a job. I do not see them as a sport but as a hobby.
The definition of the term mountaineering based on a few reputable sources I gave in my separate text where I discuss the differences between hiking and mountaineering.

Note also that in Wikipedia the term alpinism is also used as a synonym for mountaineering, and I must say I am not sure if this is completely correct.
On the other hand, the term mountain climbing is practically always used as a synonym for mountaineering. In my view this is indeed so, and I would not make any difference between them.
Surprisingly, in one source I see that the author is trying to make a difference, see what he writes:

When I look at these two “definitions” they make no sense to me at all. According to this, only the hardest route to a mountain summit is mountaineering while other routes are not.
Frankly speaking, I would rather reverse these two definitions, but this is pointless in any case. So I would not discuss this any longer, it is better to provide some answers to the questions that I have seen around.
What are the benefits of mountain climbing?
There are many ways to approach this and to find good sides of mountain climbing. This is an activity that involves your complete body so this is an excellent full body workout. It is also good for your self esteem, and you can start it at any age as I discussed in my separate text.
You can make this your profession, and you can also generate income from this activity as I discussed in my text about hiking, there is no difference in this respect.
What are the disadvantages of mountaineering?
Actually there are no disadvantages except for possible hazards. However, this depends on how you approach mountains and what you do. I have been mountaineering for several decades already, so far so good, and I have never had any accident.
Here is a video with quite a number of questions about mountaineering answered, have a look:
In summary, mountaineering and mountain climbing are the terms used for the same sort of outdoor activity. This is something most of people do for pleasure, but for some this can also be a full time job.
Thank you reading. For more texts of this type please check under the category Mountaineering here in the site. See my text on how expensive mountaineering is.
Bookmark this site and keep as a reference, you will always find new texts added here, with answers to questions that people ask about outdoor activities.