Mineral acids are inorganic. They originate from minerals, and were synthesised by mineral reaction. Hence, they are called mineral acids.
Organic acids on other side are composed primarily of Carbon, Hydrogen Nitrogen and Oxygen . They are organic, ie, seen commonly inside living cells or tissues.
How both are different:
Mineral acids have ionic bond. That means, mineral acids have a well defined positive and negative entities, which dissociates on addition of polar solvents like water.
Eg : HCl had H+ and Cl-and bond is ionic.
Organic acids have covalent bond. The bonds are polar, so they do release H as H +, however, its still covalent.
Mineral acids dissociates completely in polar solvents due to their ionic nature of bonds. So H+ concentration is high in solution. They are immiscible in non polar solvents.
Organic acids donot dissociate completely in polar solvents due to their covalent nature. So, H+ ion concentration is low. They are soluble in inorganic solvents.
Thus, inorganic mineral acids are strong , due to more H+, while organic acids are weak.
Why the difference:
Organic acids have covalent bond.. The bond exists due to electronegativity difference.
Mineral acids have ionic bond, existing due to perfect ionisation of elements.
This is why both are different.