The recently posted article does a pretty good job of summarizing the differences.
One minor critique I have is that it says brokered CDs are callable, which is true for many but not all offerings. Another interesting point is that, according to the article, not all brokered CDs are FDIC insured. I haven't seen that yet, but good to know that those exist and to double-check before purchasing.
Another feature that the brokered CD offers different from the direct CD is the "death put"...basically giving the beneficiary the option to redeem the brokered CD at par when ownership is transferred. It's unclear to me how banks/CUs commonly deal with this issue. When the estate is settled and there is no joint account holder, do banks give the beneficiary the option to redeem the direct CD early without penalty or not? Most likely the handling of this varies according to the banking/CU policy. Perhaps those on the board who have gone through this process can weigh in on their experiences.