Exponential vs Weibull Distributions -- What is the difference

Understand the implications of using an Exponential vs a Weibull distribution to model time between failures. What is appropriate depends on what you are modeling.
If you have data try using the Metalog distribution. Real data might not actually either of these distributions. See my series of UA-cam videos on the Metalog distribution.

[@dmi3kno]

Brian,
I started looking at the world of distributions through the lens of quantile functions (inverse CDF) and statistics started to make a lot more sense. The quantile function of exponential distribution (without location and scale parameters) is Q(u)=-ln(1-u) and the quantile function of Weibull distribution is Q(u)=[-ln(1-u)]^k, so it is just a positive power transform of the exponential distribution. That's why under k=1 Weibull and Exponential are the same.
Have a look at Brad Powley's or Chris Hadlock's thesis. They both studied quantile function transformations and provide useful example tables which highlight the relationships between the distributions.
I made my little {gilchrist} package in R, which transforms simple(r) quantile functions into more complex quantile functions.

[@brianchaplin278]

Very helpful.
I was using both the Weibull and Exponential distributions to model organizational lifetimes.
If the Weibull k parameter was near 1, I could use the simpler exponential model for the survival curve. It meant I could explain the expected lifetime in terms of a half life which made it easier to explain to the client. "This might suggest random external events are causing mortality" according to Wikipedia. I'm still investigating if that's the case with my client.
Under other circumstances the Weibull k parameter was less than 1, implying the failure rate decreases over time, which is typical of a concept or organizations. en.wikipedia.org/wiki/Lindy_effect. The Lindy effect applies to "non-perishable" items, those that do not have an "unavoidable expiration date". That would apply to organization lifespans.
So the Weibull k parameter value is a clue to causality for organization lifespan hazard analysis, I think.