.380 is a popular choice for women and small boys.

Seriously, the size of the majority of .380 pistols is a good fit for a smaller hand. Medium and large hands have to really work to get a good purchase on one (flesh to firearm contact).
I compare it to trying to use a beefy full size screwdriver to a shorty screwdriver that's meant for very tight spaces. The larger one is much easier to grip for added leverage if loosening stubborn screws. The more contact it has with your hand, the more power and control you have.
.380 ammo can be difficult to find and may tend to be higher priced than comparable 9mm ammo. I believe there's a wider selection of 9mm bullet materials, weights and types than you normally find in .380 without ordering online.
A .380 may be slightly easier to conceal and lighter, depending on the specific models being compared.
A .380 is often less ergonomic making it harder to shoot accurately. YMMV based on hand size and strength.
i think a .380 would make a good backup pistol for close-up, personal encounters.
With the number of options in 9mm pistols and ammo today, i'd go with a 9mm. Not only does it make an effective CCW and HD weapon, if you decide to change up with another pistol, the 9mm will be easier to resell than a .380 IMO.
Have you ever gone to the range with a .380? if not, i suggest you do a "taste test" -- shooting a .380 and a 9mm. Try to use comparable sizes to make the test fair.
Based on experience, you're more likely to pick 9mm as your favorite.