Under the hood, RAM and SSDs use very different structures to store memory. SSDs use "flash memory". To store a bit in flash, the SSD uses a larger than usual voltage to inject and trap electrons into part of the memory cell. This is a stressful process for the silicon, so there's a limited number of times it can be done. The benefit is that the injected electrons stay put for decades.

RAM (specifically DRAM) stores charge in tiny capacitors. These don't take anything special to charge up unlike flash. However, they are very leaky and will lose their charge (and therefore the memory) in a handful of seconds. RAM chips actually read and rewrite their memory several times per second to make up for the leakyness. Because of this, RAM needs to always be powered to keep its memory. This makes DRAM unreasonable for SSDs.

There are a couple other types of memory, but they have different power and space trade-offs. One example is SRAM. It needs power to keep its memory, but it doesn't need to constantly refresh, so it doesn't take much power. It can be rewritten indefinitely. However, it takes up more space on a chip than DRAM or Flash, so it's much more expensive per byte.