The sentences “vorrei diventare dottore” and “mi piacerebbe diventare dottore” are essentially equivalent, although the latter would more likely be “mi piacerebbe fare il dottore”.
However, “vorrei” and “mi piacerebbe” are not generally equivalent: you should never translate “I'd like some tea, please” into
*mi piacerebbe del tè, per favore
that can unfortunately be found in cheap translations. It should be “(vorrei) del tè, grazie”; the verb can be omitted.
One could actually state that there is a tiny difference between “mi piacerebbe andare in Spagna” and “vorrei andare in Spagna”: the former might mean a vague desire, whereas the latter could show a stronger intention. However, in this context the difference would be so tiny that nobody could perceive it. Let's imagine a dialogue:
— Mi piacerebbe andare in Spagna.
— E quando vorresti andarci?
I don't think anybody would reply with “E quando ti piacerebbe andarci?” that would sound like the “tea” example above. Another dialogue:
— Vorrei andare in Spagna.
— In che stagione ti piacerebbe andarci?
Such a reply could emphasize the fact that there are large seasonal differences in Spain, asking what season the person would like the most. All in all, these are nuances.
The “tea” example shows that Italian politeness rules are very different from the English. For Italian, the conditional of volere is normally understood as a polite form.