ATFs are usually less viscous and designed to work under lower pressures and temperatures than hydraulic fluids. The operating temperature range of hydraulic fluids is larger i.e. they maintain their viscosity better in a broader temperature range. Also hydraulic fluids are very pure compared to ATFs. ATFs are rich in detergents. If you google 'detergents in ATF' you will find threads discussing the use of ATFs as an engine cleaner. I guess ATFs have more detergents in it as they are expected to work in more contaminated environment than the hydraulic fluids. With higher quantity of detergents ATFs can trap more contamination before they start making sludge.
ATF is very complex fluid compared to normal hydraulic fluids. The first difference is the friction properties. ATF have to allow certain friction for clutch and at the same time low friction for other components. Have to balance static coefficient of friction and dynamic coefficient of friction. Conventional hydraulic fluid use ZDDP for anti-wear properties, but ATFs contain very small quantities of zinc additives (ZDDP). ATF also have fairly high quantities of detergents. They are VI improved fluids and contain high dose of VI improvers. May lose viscosity due to shearing of VI improvers. Above all, different OEMS have different requirements and is an OEM specific fluid compared to hydraulic fluids.