mostrar - to show
Muestrame el anillo que te gusta tanto. - Show me the ring you like so much.
demostrar - to prove.
Siempre me dices que vas a hacerlo pero no lo termines nunca. Por eso, no te creo. Demuéstrame para que pueda confiarme de ti.
You always say you are going to do it, but you never finish it. That is why, I do not believe you. Prove me, so I can trust you.
I hope it was helpful. If there is anything else I can help you, feel free to ask me.
Have a nice day.