Fate and Destiny both exist.
Because the deinitions of the words have become interchanagble in modern slang they kinda have been tossed back to mythological context which sparked your debate which is real and which isn't . But they both are real and active in our everyday lives. Fate is a negative where destiny is a positive so they differ in that regard.
Fate has it's roots from the words final and fatal meaning an end or death. (ie: we are all fated to die for no mater what we do and how we live as humans our fate is the inevetable death. If i continue to spend my money carelessly my fate will be that of a bum)
Destiny is a sum of events or actions, choices, traits and pure chance to achieve a result; it is not a fatal result but always a result .
everyones fate is the same we will all die but everyones destiny is different due to ever changing factors (ie: personality, luck, behaviour and choice)
Examples of modern use for fate and destiny.
in a conflict between two combatants one shall say "meet your fate" to imply meet your inevetable death.
In a young child showing exceptional talent personality and behaviour soneome may say " you are destined to do great things"
It would not make sence for the phrase to read " meet your destiny" or " you are fated to do great things" made clearly on the basis that fate is a negative and destiny is a positive.
So you can see both fate and destiny do exist when the root meaing of the words is clear. They both consists of events and have inevetable outcomes but the inherent differnce if fate is always an end or fatal out come or result of a fatal event and destiny is a success or aspiration for sucess regardless of how in size.
With that said the argument can be made that fate is used in the context of a positive. (ie: meeting you in my time of need was fate or it was fate that led me to you)
here fate is the marker at which and event/circumstance has ended so the death in this case isn;t a literal death but rather the ending of a peroid of emotions ,events or circumstance.
Besides being opposites (negative/postive) fate and destiny oppose each other in that, fate has no future or no past, once fate has come to be it is and end or death. Fate is a final result. Where destiny has a past, present and future. there is life after destiny. ( Ie: (past)..... looking back now at my trophies, my desity was to have been the greatest race car drive in the world. (present)..... this was my destiny to be the great race car driver i am today. (future)..... this is my destiny, to become the greates race car driver the world has seen.