Magnets are iron that has been magnetized through modern industrial processes. That is, it is iron with magnetism, but the magnetism is not stable and will demagnetize over a long period of time.
Lodestones are magnetic stones in nature, that is, permanent magnets. From the perspective of traditional Chinese medicine, lodestones have the function of regulating the human body and are usually used for health care. Many insoles, bras, and pillows still use lodestones, but they do not cure diseases.
Magnets can attract metals such as iron, nickel, and cobalt and are commonly known as "magnetic stones for attracting iron". They can be divided into ordinary permanent magnets and electromagnets that only have magnetism when electricity is passed through them.
What we usually call "magnetic stones for attracting iron" is actually called magnets. But the real "magnetic stones for attracting iron" is a kind of ore that contains a large amount of magnetic substances and can attract iron, so it is called "magnetic stones for attracting iron".
Because iron can be magnetized (that is, the constituent atoms of iron are rearranged), it has the ability to attract iron like a lodestone. But its magnetism is not as strong as that of a lodestone.
Nowadays, both terms are used for magnetic devices, and there is not much difference between them in detail.