I agree with Rusty and with your original definitions.
"Sobresalir" implies that something sticks out in a comparison, where there is something else taken as a reference:
- Los brotes de la planta sobresalen ya de la maceta. -> The sprouts of the plant are sticking out of the pot.
- Juan es un estudiante sobresaliente. -> Juan is better than his average classmates.
- Yo no sobresalgo en la calle. -> I'm like any average person in the street.
- La cola del gato sobresale por encima de los arbustos. -> I can see the cat's tail sticking out of the bushes.
- Nuestro gobernador sobresale entre todos por incapaz. -> Our governor stands out from the others because of his helplessness.
And "asomar" is mostly used implying some sort of movement in order to have a glimpse:
- Los brotes de la planta ya asoman por encima de la tierra. -> I can see the plant sprouts coming out of the soil.
- No asomes la cabeza por la ventanilla. -> Don't lean out of the car window.
- Me asomé, pero no vi a nadie. -> I came around to look, but nobody was there.
- ¿Quién es ése que se asoma por la ventana? -> Who's that guy peeping through the window?