Противоположное значение (opposite meaning) слову (word) твердый (hard) - мягкий (soft) Противоположное значение слову жесткий (stiff or rigid) - гибкий (flexible) Твердость это свойство материала. Не зависит от конструкции. Hardness is a material property.It does not depend on the a structure. Примеры (examples): Сталь тверже резины (Steel harder rubber.). мягкая подушка. soft pillow. Жесткость это характеристика конструкции. Жесткость зависит и от геометрии конструкции и от используемых материалов. stiffness is a characteristic of the structure. The stiffness depends on the geometry of the structure and the materials which have been used. Примеры (examples): Жёсткая конструкция Stiff structure (or flexible structure). гибкий шланг.. a flexible hose. from wiki: Hardness is a measure of how resistant solid matter is to various kinds of permanent shape change when a compressive force is applied. Some materials, such as metal, are harder than others. Stiffness is the rigidity of an object — the extent to which it resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Stiffness is defined as resistance towards deformation.

твёрдый -- the quality of the material, for example: Подушка мягкая, а стул твёрдый. жёсткий -- the opinion of the material Эта подушка, когда-то такая мягкая, стала жёсткой.

The literal meaning of these two words is very different. Твердый is the opposite of soft - мягкий. Жесткий is the opposite of smooth - гладкий. Твердое вещество//жесткая подушка. Figurative meaning of these words isn't so different. Твердый means determined, resolved, жесткий - determined, resolved and quite abrasive

ТВЁРДЫЙ - hard, as in "not soft", "not liquid". The one that has a hard (to the touch) surface - "firm" in figurative and sometimes literal meaning - in mechanics, "rigid body" is "твёрдое тело" - in physics "solid" state is "твёрдое" состояние ЖЁСТКИЙ - rigid, not flexible, hard. - "firm", "hard" when talking about supposedly "soft" items like pillows, mattresses, armchairs, beds, sofas - about structures: rigid, inflexible structure. The one that stands pressure from outside well. - physics: "hard radiation" = "жёсткое излучение" - figurative meaning: harsh, strict. When talking about a person and their decisions, "твёрдый" is more about being firm and steadfast. "Жёсткий" is about being strict, rigorous and harsh (i.e. possibly unpleasant).

По большому счёту, разницы почти нет. Но, в зависимости от контекста, предпочтительнее либо первый вариант, либо второй.