1 - To use DRAM, it requries a DRAM controller to refresh the DRAM. The controller takes up silicon space on the dice, which can raise the cost of the microcontrollers.

2 - Most embedded applications don't need crazy amounts of RAM, thus is why you don't see a DRAM Controller on them. Unfortunately if you need large amounts of RAM, most microcontrollers are immediately crossed off the list because they don't have a DRAM controller. I wish there were some popular microcontrollers with 8 to 32MB of DRAM inside the same chip and the more the better.

3 - DRAM can be a pain in the ass to initialize / setup / tweak, and you might have to make software changes to support alternate DRAM chips if you can't source older memory.

4 - You can't use DRAM without first configuring it and the DRAM controller. With SRAM, technically you could use JTAG in the microcontroller to download a program into SRAM, then immediately boot from the SRAM, which can be very helpful during development phases.

5 - If you stop the clock or DRAM controller during debug, then you lose everything. The best microcontrollers don't have issues, but some have weird problems during debug, so user beware.