“What's the difference between a stammer and stutter?”

They are interchangeable. Stammer is typically the British term for stuttering.

Really, it's the root cause of the problem. Stammer is really more of a layman's term for a fluency disorder, whereas stuttering is one type of fluency disorder. Stuttering is characterized by repetitions, blocks and fillers. Rate of speech can at times be a factor but not always. Other types of fluency disorders such as cluttering are often times misdiagnosed, as cluttering has some of the same characterizes as stuttering. One big difference is usually individuals that clutter, aren't aware that they are doing so. Remediation of cluttering is more difficult and almost always demonstrates a fast rate of speech. Typically, individuals that clutter do not have blocks or physical concomitants unless the individual also stutters.

It depends on who you are speaking to and in what country. In the U.S., I was taught they are used interchangeably and are not much different. In Britain, I was taught it depends on the location the person gets stuck on. I learned from different people to basically look at it as anyone can stammer on a word, get stuck on it and try to repeat it and flow through it, but then there are persons who stutter. There is an entire other component to being a person who stutters. The National Stuttering Association is a good reference for you to get more clarity. I treat persons who stutter and persons who clutter but I've learned not to get boggled down in the dynamics of stammer vs. stutter because I've met people who stammer but don't stutter, and people who stutter that stammer. I just know it when I hear it and it gets complicated trying to explain it without being with you in person. Hope this helps a little.

Stuttering and stammering are essentially the same thing. "Stammering" is more commonly used in the UK and Canada, while "stuttering" is more commonly used in the US.