Those are prepositional phrases and not dependent clauses.
You're asking for the difference between these prepositional phrases:
With the growing of grass, ....
With growing of grass, .....
With the growth of grass, ....
The only way to do this is to finish them by adding a clause.
1.) With the growing of grass, the landscape will become beautiful.
2.) With growing of grass, the landscape will become beautiful.
3.) With the growth of grass, the landscape will become beautiful.
Now that a clause has been added I'll try to give the roles of the prepositional phrases using, to the best of my abilities, Kenneth Burke's Grammar of Motives.
1.) With the growing of grass, the landscape will become beautiful.
The definite article "the" in this first one indicates that the speaker and listener have an shared agreement or shared understanding that grass is or will be or has been planted. This planting of grass is done or was done purposefully with the intent of making a change of the landscape. In this case, the purpose was or is to make the landscape more attractive.
This does not entirely mean that the definitely article could mean that the grass was done unintentionally or accidentally; accidental or unintentional grass growing is quite possible. But the use of the definite article "the" attaches some meaning of understanding. It may even attach what type of grass is being grown, intentionally or unintentionally.
At this point I would look at the meaning of "the" when attached to nouns and noun phrases.
When someone says "the dog bit me." It implies, sometimes subconsciously, that the listener and speaker have a shared understanding of what particular or specific dog is being talked about.
But it is also quite possible that the speaker and listener have different dogs in mind. The speaker maybe talking about their pet dog, Max but the listener, who may believe that their own pet dog could never hurt them, could assume it was the neighbor's dog.
Now we need to look at the gerund "growing" or any verb that ends with
[-ing]. In this case I'll stick with "growing" because it's most relevant.
The use of "growing" in this sentence slips very subtly I'm our minds a progression of grass growing that the speaker and listener will witness on a recurring basis, like daily or maybe hourly(some grass does grow that fast). So the use of "growing" imbues in us an ongoing image of constant change or motion.
2.) With growing of grass, the landscape will become beautiful.
Here, the definite article is absent. But the semantics involved in sentence number one(above) still remains, but the sentence takes on a cultural aspect and thus can be described as more colloquial than formal.
If someone from the south was walking in a library and a book happened to fall on his head, he might say "damn book fell on me!"
Here an article is missing, but we know he means "A damn book fell on me!" Which is formal way of saying "damn book fell on me!"
3.) With the growth of grass, the landscape will become beautiful.
Here, the key different is growth.
One of the first definitions of "growth" in the Merriam Webster dictionary is "a stage of the process of growing."
Because the landscape is expecting to be beautiful, we can assume that the stage which the speaker is referring to is the final stage, where grass my cover nearly every foot of the landscape and is just about complete. It cannot mean when the grass has grown halfway in covering the landscape because it match the finality of the setting, which is a beautiful landscape.
I guess it's a good time to mention that when I say growing or growth of grass, I mean the spread and covering of grass, and not the height to which grass can grow.
The use of "growth" also seems to carry the idea that little attention to be paid to the process of it growing and the speaker and listener have little interest in any stage but the final stage, which is grown grass making the landscape beautiful.
Keep in mind that the definite article I also present here, so I would refer you to my explanation of number one.