They are different or the same, depending.
Marketers of one other the other often label them as one or the other with the other in parenthesis.
They are both the same species.
In the US both words are used interchangeably for either bean. I suppose in Italy they may only recognize a handful of related cultivars as being a true garbanzo.
There are literally 100's of thousands of cultivars of Cicer arietinum. From tiny quick cooking ones (15 minutes without soaking first) to gigantic long-cooking ones (1.5 to 2.5 hours, the only ones sold is US supermarkets), to every size in-between and cooking time in between. There are also purple ones, orange ones, black ones, brown ones, mottled ones, near-white ones, green-ripe ones, and kinds that will pop like popcorn.