Although similar, there are some minor differences.
As you mentioned, the safety on the Browning is slide mounted versus the frame mounted Beretta safety. Beretta factory magazines also held an additional round for a total of 13. I believe all BDAs came with the rounded trigger guard while only some early Beretta "Cheetahs" came with a rounded trigger guard, later opting for the beretta "FS" style as shown in these pictures, which also depict the cutaway slide of the Beretta. The sight shape also varies between the two, with the Beretta offering more of a flat sight surface facing you, which might make it easier to spot.
They're not clones, strictly speaking, but they're definitely cousins. FNH contracted for the BDA and a lot of the pictures that I found online of the right side of the BDA are marked with the FNH roll mark as well as the Beretta "PB" roll mark and "Made In Italy."
The fact that the 84 is still being produced might make it easier to get replacement parts for than the BDA, which has been out of production for a decade and a half (except for a few custom runs).
Browning BDA 380Bereta 84 | Beretta 84 | Browning BDA |
Production Years | 1977- | 1977-1997 |
Weight | 1.46lbs | 1.41lbs |
OAL | 6.8" | 6.8" |
Barrel Length | 3.8" | 3.8" |
Capacity | 13 | 12 |
Safety | Frame | Slide |