Our "higher reasoning" and "state of awareness" (needs defining) gives us the ability to do thigns other animals can't. For example chronic pain sufferers are taught how to manage their pain with a variety of CBT techniques. Not something you can teach a dog or cat.
People in intense periods of intense suffering may have thw ability to dissociate from the experience ("go to their happy place") to lessen the pain experience.
We're not aware animal shave this ability.
If anything mammals of all kinds that feel pain don't have our higher cognitive ability to help manage and supress it.
Having said that it's possible we feel more emotionally complex pain. Pain induced from our own minds by remembering trauma or imagining painful situations. As someone pointed out below a dying animal probably isn't thinking about the loss of it's family as it's dying. But it will be feeling the pain of dying acutely.
I have been stung by bees/wasps several times, and the times when I was youngest and the least self aware were the worst. I was in absolute screaming agony as a small child. Then one day as an adult I was startled to be stung and found the experience to be completely different. Sure, it hurt and was really sharp like someone just jabbed a needle into me, but my response was to laugh, not cry. I also have the capacity to just not give a fuck (I recognize the cause of the pain isn't going to kill me) when I'm in a fair bit of pain and just do something else (provided I can still physically move, which isn't always a given) and this is helpful for tuning it out.
So from my personal experience, I would say absolutely: animals have it worse, not better.