Hope you are doing great! :)
The word "vorrebbe" comes from the verb "volere" (to want) while the word "piacerebbe" comes from the verb "piacere" (to like).
In certain cases, there is just a slight difference between them in terms of meaning.
Let's see an example:
ITA: Lei vorrebbe che andassimo là - ENG (literally): She would want (us) to go there
ITA: A lei piacerebbe che andassimo là - ENG (literally): She would like (us) to go there
In both cases, her desire is for us to go there. Both in Italian and English the meaning is very similar.
However, in other cases, the meaning changes slightly, let's see another example:
ITA: Vorrei andare in Spagna - ENG (literally): I would want to go to Spain
ITA: Mi piacerebbe andare in Spagna - ENG (literally): I would like to go to Spain
The first sentence (vorrei) conveys a more definite plan that might actually become true. The second sentence (mi piacerebbe) conveys a more vague desire, something that you would like but perhaps you haven't planned when to do it yet.
Depending on the context, you will use the appropriate verb, for instance:
ENG: I would like to go to the toilet - ITA: Vorrei andare al bagno
ENG: I would like you to complete your homework - ITA: Vorrei che tu finissi i compiti
ENG: I would like to become an actress - ITA: Mi piacerebbe diventare un'attrice
ENG: I would like to try Indian food - ITA: Mi piacerebbe provare il cibo indiano
As you can see, the tendency is that the verb "volere" is used when you have a more definite plan, while the verb "piacere" is used when you have a desire.
Let me know if this helped! :)