You could call the construction of the sentence in question the
"on the one hand, but on the other hand"
type of question.
On the one hand, I would like to be an Olympic athlete, but on the other hand, I am not willing to undergo the vigorous training needed to become one.
The but in the above sentence functions to conjoin the two independent clauses, but it also functions to separate the opposing ideas. One of the ideas you could call theoretical (viz., "I would like to be . . ."), and the other one realistic.
I would like to be rich, but I will probably always be poor.
Here again, the theoretical clause (key word, would) is in opposition to the realistic clause. Safe to say, most people would enjoy being rich, at least in theory, but thinking realistically they know in their hearts they will always be relatively poor, as least compared to a truly rich person, such as a billionaire!
For further helpful information on the various ways of how to use the auxiliary verb would, look here.
In summary, conjunctions such as the word but (and other conjunctions such as or, and, because, and as) join together two independent clauses. When but is the conjunction, the construction of the complex sentence indicates a contrast between two sentences.
One final example,
On the one hand, I would like to correct her grammar, but on the other hand, I am afraid she will be offended if I do.