The children are eager to start the novel.
The children are eager to begin the novel.
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Begin, when used as transitive verb, means "start, perform, or undergo the first part of an action or activity."
Start, when used as transitive verb, means "cause (an event or process) to happen", or "cause or enable someone or something to begin doing or pursuing something."
In "the children are eager to start reading the novel," start means "embark on a continuing action."
"Begin" might also be for something that has already started. But to "start" marks the actual/exact time of launching an activity (to understand more clearly, consider these two examples: This is just the beginning [meaning, all the initial period] . It's 10:00 o'clock, folks; let's get started [whenever we talk about a specific time, we rather use the word "start"]. Since we are bound by the two sentences above, we cannot speak of the noun-verb possibility of each word as a difference.
In the context of your posted sentence, there is little difference between the use of the two verbs. Most readers would assume the sentences mean the same thing. In a different context, there is a potential ambiguity in both usages associated with the missing verb:
I must begin the novel.
Do I mean?:
I must begin reading the novel.
I must begin writing the novel.