'It's my ....' means 'This is mine'. Often we'd say this in response to the question 'Whose ... is this?' or 'Is this your ... or your brother's?'. It means that this thing belongs to me, rather than to someone else.
'It's a .... of mine' means 'I have several ....s, and this is one of them.'
They are often interchangeable, but the focus is slightly different. Which one you choose would depend on the context.