They both tell a story. Whether it's a brief moment in flash fiction or a more developed narrative in a short story, there's a story being told. Also, they both can have a beginning, middle, and end, even if in flash fiction it's very condensed.
One of the main similarities is that both flash fiction and short stories require a certain level of craftsmanship. The writer has to carefully choose words, structure the piece, and develop the narrative arc. In a short story, the writer has more words at their disposal to build this, but in flash fiction, it's about being extremely economical with words while still achieving the same effect. They also both share the aspect of having a conflict or tension. This could be a conflict between characters, within a character, or between a character and their environment. In a short story, this conflict can be explored in more detail, while in flash fiction, it's presented in a more concentrated form.
Well, both use language creatively. They have to, in order to make an impact in a relatively short amount of space. Flash fiction and short stories also both rely on the author's ability to create vivid images in the reader's mind. They might do this through detailed descriptions in a short story or more spare but evocative language in flash fiction. Another thing is that they can both be about any topic, from the ordinary to the extraordinary.