For example, "我不能睁开我的眼睛" and "我睁不开我的眼睛" (不 obviously being the negation of the 得 potential form). What is the difference in nuance between these two sentences?
The opposite of '我睁不开我的眼睛'(I can't open my eyes) is '我睁得开我的眼睛'(I can open my eyes)
[不] and [得] are [potential particles]. They have to be placed after the verb.
For example: 去得到/去不到
去(go) is the verb; 得/不(can/cannot) is the potential particle; 到(arrive) is the resultative particle
The opposite of "我不能睁开我的眼睛" (I can't open my eyes) is '我能睁开我的眼睛'(I can open my eyes)
[能] and [不能] are [auxiliary verbs] (can; able to/ can't; unable to). They have to be placed before the main verb
For example: 能去 / 不能去
能(able to/can) is the auxiliary verb; 去(go) is the main verb
The 得／不 potential form is used with result verbs. As in your example:
我睁不开眼睛 ---> 开 (open, separated) is the result of 睁
Therefore the potential form tends to express an objective ability or inability to do something. Eg:
我打不开门 I can't open the door (it is too heavy, blocked, broken...)
我睡不着 I can't sleep (too anxious, there's noise...)
今晚我做不了饭 Tonight I can't make dinner (I'm too tired)
The 能 form sounds more like a subjective ability. It is used with generic actions that don't have a result (in the sense of a result complement), polite requests, to express moral/legal constraints. Eg:
你能想象吗 Can you imagine it? (subjective non-physical ability)
你能帮我忙吗 Can you give me a hand? (request)
今晚我不能做饭 Tonight I can't make dinner (generic action)
我丢了驾照，我不能开车 I lost my driver's license, I can't drive.