The way you phrased that question made me do some real self reflection. I can say confidently that I'm proud to be a stutterer, but I really had to to dig deep to decide if I was proud of my stutter. I'm not even sure if I can explain why I think there's a difference between the two. But after really thinking about it, I can say that I am proud of my stutter.
I think that I am a better person than I'd be if I were fluent. I think I have more patience and compassion. I think I try to listen to people carefully because I expect that in return. I'm also proud of my stutter because it makes my voice more unique. The only times I know I'm saying what I want is when I'm stuttering. If I was told I could have fluent speech, I wouldn't want it.
For me, most of the hard parts about stuttering pride are external. Being a proud stutterer means I talk a lot more than I used to. Just like with being gay, you don't just come out once. Every time I talk I reveal myself as a stutterer. No matter how proud I am, I still can't be sure if others will treat me fairly. That can be frustrating and sometimes discouraging.