The biases are obvious. If you like one president and hate the other, then you are inclined to cut more slack to one of these celebrities than the other. That kind of bias is relatively obvious, and the partisan and ideological lenses that people use get referenced a lot here. There is another level here, though, which is the question of the extent to which these are actually different, and the extent to which that matters.
Threat. I comment here about language, largely because I am bothered by the extent to which others are bothered, pointlessly, by language. "Shit, piss, cunt, fuck, cocksucker, motherfucker, tits." George Carlin's seven dirty words. Granted, I am most cautious about the use of the word, "cunt," than any other word on that list, but any fucking asshole who wants to fucking tell me I can't say fuck is a shithead motherfucker who pisses me off. Tits.
Did I do anything wrong there? Yes. I ended a sentence with a preposition, and "Tits" wasn't a complete sentence. I feel ashamed.
On the other hand, I won't use racial slurs, and I sort of ducked the use of the word, "cunt." I also didn't include "cocksucker" in that sentence. Why? It could come off as homophobic in that context, just as "cunt" can come off as misogynistic. They don't quite rise to the level of the n-word, but built into all three, to some extent, is a history of violence against African-Americans, women and gay people. I'm a straight, white male. Anyone who claims not to understand why a white person can't say the n-word, but in certain circumstances, a black person can is being intentionally obtuse. It is about whether or not there is an implied threat of violence, however subtle. If I say it, it's gonna be there. I can't escape history, and I have no desire to try.
And that brings us back to violent imagery and presidents. We have had four presidents assassinated, with attempts planned and made on more. Any words or imagery at all involving violence towards the president conveys that history.
Are Nugent and Griffin the same in that respect, though? Um, no. Nugent is a hard-core gun advocate. When he says, "suck on my machine gun," he actually owns lots of machine guns, advocates everyone owning lots of machine guns, and belongs to a movement that, throughout Obama's Presidency, regularly touted that line from Jefferson about the tree of liberty and the blood of tyrants. His fans are drunken dudebros, similarly armed and similarly "minded."
Kathy Griffin is a comedian. There is probably cutlery in her kitchen capable of severing a head, as there is in any properly acquitted kitchen, but I doubt she is a member of "Knife Rights" (yes, it is a real group), and nothing in her history, general tone or fan-base has the same kind of violent, angry theme as Nugent.
Griffin's and Nugent's actions are not equivalent because they do not contain the same level of threat. That said, any violent imagery regarding the president has some level of threat. That is unavoidable given the history of presidents being assassinated, and assassination attempts. No white person, no matter how jokingly, can say the n-word without the baggage of history, and no comedian, no matter how she thinks of herself, can post that image, without the history of presidential assassinations. That doesn't make it equivalent to Nugent, though. Nugent, let's remember, also peppered his language about Obama with gems like calling him a sub-human mongrel, and I could go on, but there is no need.
How we react to Griffin is difficult to separate from how we think of Trump, just as how we reacted to Nugent was difficult to separate from how we thought of Obama. It is important to make that separation. It is also important, though, not to equate Griffin and Nugent. It always comes back to the level of threat implied. With Nugent, the level of threat was higher.