Falco vs. falcone
In Italian, both falco and falcone translate to falcon. However, they are not perfectly interchangeable. The primary difference is that falco is the general, common term, while falcone is an augmentative form used in specific contexts, most notably falconry, to refer to a large or female falcon.
Falco
A2This is the standard, generic, and most common word for falcon. It is used in zoology and everyday language to refer to the bird of prey in general, regardless of its size or sex. It can also be used metaphorically to mean a hawk in a political or financial context, or to describe someone with sharp eyesight.
Un falco pellegrino può raggiungere velocità incredibili in picchiata.
(A peregrine falcon can reach incredible speeds in a dive.)
Dal sentiero abbiamo avvistato un falco che volteggiava sopra le cime degli alberi.
(From the trail, we spotted a falcon circling above the treetops.)
Il ministro delle finanze è considerato un falco per le sue politiche rigorose.
(The finance minister is considered a hawk for his stringent policies.)
Mio nonno ha la vista di un falco, riesce a leggere senza occhiali.
(My grandfather has the eyes of a hawk, he can read without glasses.)
I falchi sono predatori magnifici e molto efficienti.
(Falcons are magnificent and very efficient predators.)
Falcone
B2This is the augmentative form of falco (the -one suffix implies largeness). Its main, technical use is in the world of falconry, where falcone specifically refers to the female falcon, which is notably larger and more powerful than the male. By extension, it can be used outside of falconry to emphasize the impressive size or majesty of a particular falcon.
Il falconiere tiene il suo falcone sul pugno guantato.
(The falconer holds his falcon on his gloved fist.)
Nella caccia con i rapaci, il falcone è apprezzato per la sua stazza e forza.
(In hunting with birds of prey, the female falcon is prized for her size and strength.)
Quell'uccello maestoso che hai visto non era un gheppio, era un vero falcone.
(That majestic bird you saw was not a kestrel, it was a true large falcon.)
L'addestramento di un falcone richiede grande dedizione e competenza.
(The training of a large falcon requires great dedication and skill.)
Un enorme falcone si posò sulla roccia, dominando la vallata con lo sguardo.
(A huge falcon landed on the rock, dominating the valley with its gaze.)
Summary
In essence, use falco as the default, everyday word for any falcon. Use falcone when you are specifically talking about the larger female falcon within the context of falconry, or when you want to consciously describe a falcon as being particularly large and impressive.