Is there any difference in meaning between the two following sentences? Are "quando ero X" and "da X" both usual when they express a past time period?
Quando ero un bambino, piangevo molto.
Da bambino piangevo molto.
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The two phrases have the same meaning (as @egreg pointed out in his comment); the latter one seems more informal, but this can be considered my personal opinion.
Please note that you can use "Da + (noun/adj)..." also in other sentences like:
Da grande farò il pilota di Formula 1
= When I grow up, I will be a Formula 1 driver
Te lo dico da amico...
= I tell you this as a friend...
You could say that "Da + (noun/adj)..." has a meaning like "in the condition of..."
As mentioned in other answers, there is no difference when you are referring to yourself. However, you cannot use the da+... construction to refer to others without changing the meaning.
For example, "Da bambino, Sandro Pertini era Presidente" means that Pertini was president while he was a child, while "Quando ero bambino..." means that he was president when YOU were a child.