I haven't done either yet, but this is how I've been thinking about their advantages/ disadvantages. We chose to do one DAH:
The CLFP disadvantages - Your regular 3 FPs can't overlap with the CLFP, which will make each of your CLFP days rather long and scheduled. And the CLFP can't be modified by you - only by the DSS CMs. To me, that makes them unmodifiable (they don't refresh). You can't use refresh if you want to change plans, move them up or move them to later. I have some FPs that were made by CMs that are just like The CLFP and I don't love that they can't be modified. With a CL stay, you may want to spend more time in the resort, but this basically locks you for 6 hours in the parks. I think you would be missing some relaxation resort time.
The CLFP advantages - You don't have tiers and having CL FPs means you are guaranteed to have just a short wait for each CL FP ride, even the most popular rides. With 3 days of CL FPs, that's 9 *additional* rides, non-tiered, and you can use your CLFP among *all* the parks. You can schedule your FPs for times that work for your family (for example, afternoon and early evening), instead of the specific 2.5-3 hour window that DAH is happening. You can schedule them for any day.
The DAH disadvantages - Each park has a separate DAH. If you are doing all the offered DAH, it's going to add up way more than CLFP. DAH is pretty late. For MK, it's 10 pm - 1 am, but it may take a while for day guests to clear out of the queue. This means you may not start seeing short waits until later in the evening for the popular rides. Some have indicated that 7DMT and such will still have 20ish minute waits the entire evening. If you are not a night owl, the evening timing may not work. If you had some light appetizers at 5 pm in the CL lounge, you will likely need a full dinner around 8 pm to make it through, which may need to be factored into the costs. Also, DAH is not every night. You would need to make it work with the day of the week that the DAH is offered.
The DAH advantages - Specific to CL stays, it will allow you to enjoy your CL stay because you won't be tied to FP schedules in the park. It will allow you to ride a bunch of rides with short waits in just a few hours. It includes popcorn, ice cream and soda. The night atmosphere is nice.
In the end, we chose to do one DAH on my DD's actual birthday, as the *special* birthday thing, mostly for the night atmosphere and the chance to do a bunch of rides in a short period of time.