There is more weight loss to be had switching from Diet Coke to water. It's obviously not the calorie content, but the insulin response from sweet tasting things that will occur even with artificial sweeteners.
I'm not sure about insulin resistance, but sweet-craving builds a tolerance kind of like spicy foods. That is, the more you eat sweet things, the more sweetness your body craves.
I've been using erythritol and stevia as sweeteners for years, and even with those, I find myself craving sweet things after consuming them more than when I go relatively long periods of time without eating sugary things or using sweetener.
The end result, at least for me, is that I crave sweet food more even when I use artificial sweeteners, making it harder to say no to eating shitty food later on.
I haven't read that it affects insulin response. But I have heard that it can cause you to crave more calories because it gives you the perception of consuming something but without the calories. I'm not sure what the mechanism is here or if maybe it is just psychological.
People (probably those with controlling interests in sugar cane/beet industry) have been trying unsuccessfully to find/prove any sort of real deleterious effect from aspartame specifically, and lower-calory sweeteners in general for about 50 years now, so I think we can say they're fine.
I have yet to find a single study confirming any rumored ill effects of artificial sweeteners, including aspartame. At least in any reasonable consumption amounts since you need so little of the sweetener.