What is the difference between the following two sentences? Do they both mean the same?
Why don't you start from the beginning?
Why don't you begin from the starting?
Your first sentence is fine. Your second sentence is not.
Begin and start are both verbs, so they can be interchanged.
The equivalent noun of begin is beginning.
The equivalent noun of start is start
(Beginning and start are synonyms and can be interchanged.)
So your second sentence should be:
Why don't you begin from the start?
That is, beginning and start are both "regular" or autonomous nouns. They function the same. They have their own dictionary entry as nouns. The fact that beginning ends in -ing just indicates a different formation. Compare: building. No one doubts that building is a noun!
(Nouns such as beginning and building were evidently made from the verb + -ing centuries ago, but they long since began to be autonomous nouns with all the features of other nouns, such as start (for example, they have plural forms (beginnings, buildings, starts) and they are modified by adjectives. They have their own dictionary entry as nouns.
However, starting is a gerund. It is the -ing form of a verb acting as a noun. As such it has properties of a verb, not a noun. For example it can take a direct object, it is modified by an adverb. It is not a permanent or autonomous noun. Look up starting up in a dictionary: it will redirect you to the verb start. (At least the six dictionaries I tried did this.) This indicates that starting is not considered an autonomous noun (rather, start is).
We don't say Begin at the starting. We say Begin at the starting point Thus we use starting as an adjective. It is also a gerund, as in "Starting the race from the beginning is a good idea." Notice that starting can have a direct object, when it is a gerund.
Now, to slightly complicate matters, since both begin and build are verbs, they can also be used as gerunds: beginning and building. But these words as gerunds can be thought of as different words from the autonomous nouns beginning and building (discussed above) As gerunds, they act the same as the gerund starting:
Beginning (gerund) at the beginning (noun), she read all the letters.
Building (gerund) a large building (noun) takes time.
But we just don't say:
So starting works as a gerund, but it is not an actual noun, (or a regular noun, or an autonomous noun, or a noun proper--take your pick of terminology).
One can say
In/at the beginning (noun), God created...
but we don't say:
We could say
In/at the start (noun), God created...
The first sentence is the more common usage, although technically both have the same meaning. The first sentence is a very common phrase, however, to the extent that most people hearing the second would probably assume that it was an intentional "twist" on the more common phrase.
"Start" and "begin" can mostly interchange as verbs, but "starting" is not often used in the same sense as "beginning". Theoretically, they are the same to the extent that they are both gerunds formed from verbs that are near synonyms. However, "starting" is more often used in phrases: "starting up", "starting line", "starting place", "starting point", "at starting". I don't know that there is any theory to explain why; it's probably just the way the usage has developed.
As has been stated, "start from the beginning" is a common idiom used as an instruction to describe a series of events in chronological order, from first cause onwards.
Begin from the starting, however, can only make sense if a "starting" is being used a noun, rather than a gerund.
If an engineer is explaining how an electric motor works to his students, an electric motor has two modes: starting, and running. Once you understand the starting, you can understand the running without much additional information, so typically, a teacher will arrange his notes to begin from the starting, and then work on to the continued running.
But while in theory it could be used as a sentence... Google n-grams show this word has almost no use, and searching Google books shows those words used together only when starting is used to modify a subsequent word: starting-point, starting premise, starting condition, starting block.