The difference in meaning is strictly as you've found: cerebro is the organ and seso is the tissue it's made of. But then you have figurative meanings and things get complicated fast.
Cerebro is mostly a countable noun, i.e. you say un cerebro, dos cerebros, etc. If you have a collection of preserved brains in jars, you say
Tengo una colección de cerebros.
Figuratively you can use cerebros to mean "minds" in the sense of "thinking people", so you can say e.g.
Los cerebros de la operación fueron Alice y Bob.
meaning "the minds of (=behind) the operation were Alice and Bob". This is more or less a fixed phrase or idiom.
You can use cerebro as a mass noun, but that's not really common.
Seso works both as a countable and a mass noun. If you go to a butcher's and you want brains, you ask for sesos, not cerebros. You'd say e.g.
Quiero un kilo de sesos.
meaning "I want a kilogram of brains".
Some people would use seso as a mass noun in this meaning (Quiero un kilo de seso.) That's OK too.
If someone comits suicide by blowing his brains out, you say:
Se voló la tapa de los sesos.
meaning literally "He blew off the lid of his brains" (this is also a fixed phrase).
Both seso and cerebro can figuratively mean "intellectual power", but for the meaning intended in "use your brains", you would only use either el cerebro or la cabeza. There are some idioms that employ this meaning, though:
- tener poco seso means "to be dumb, to be unthinking"
- sesudo is an adjective meaning "sensible" or "wise", but also "cerebral (as in an academic paper)"
Note that the English word cerebral is not easily translatable in Spanish. The Spanish word cerebral only means "of or related to the brain" (though some of the English meaning has crept up on it lately).