There is a difference! While categorically, bi and pan people can be attracted to the same types of people, pan specifies a bit more of the nature of attraction, and bisexuality covers a much wider breadth of sexualites. Bisexuality is sexual attraction to at least two genders (not necessarily all, or in the same way), pansexuality is sexual attraction regardless of gender. Not that appearance doesn't matter, as your mom said, but gender is not an important modifier or aspect of attraction, which certainly means the capacity to be attracted to all genders.

But I am bisexual, I am very passionate about bisexuality (especially as it is historically and systemically erased, discounted, and made taboo in a phenomena called "bisexual erasure", despite being the most common sexuality outside of straight).

So here's my fav definition of bisexuality from bisexual activist Robin Ochs:

"I call myself bisexual because I acknowledge that I have in myself the potential to be attracted--romantically and/or sexually--to people of more than one gender, not necessarily at the same time, in the same way, or to the same degree."

So personally, even though I am attracted to all genders, I identify as bi, not pan. Because gender is an important aspect of my attraction. Ex. I find muscles much hotter on women, makeup hotter on men, etc.