Why is the normal distribution used in linear models, but in generalized linear models (GLMs) the exponential distribution is used?
In GLM's the exponential family of distributions (not the exponential distribution, https://en.wikipedia.org/wiki/Exponential_family) is used to model various outcomes, Gaussian (or normal) distribution for a real continuous variable, Gamma distribution for a real positive continuous variable, binomial distribution for a discrete variable, and so on.
An ordinary linear model is just one among the selection of distributions, which uses the normal distribution.
An ordinary linear model -- which uses the normal distribution -- is just one GLM used for one purpose; other purposes suggest other distributions. Not all generalised linear models (GLMs) use an exponential distribution.
Terminology is also confusing: "Exponential distribution" in the context of a statement like 'Generalised Linear Models use exponential distributions' means "one of the Exponential family of distributions, not "the exponential distribution.
|linear response||all reals||continuous||Gaussian (normal)|
|scale parameter||positive reals||continuous||Gamma|
|Binary outcome count||integers||discrete||Binomial|
... and so on; see the GLM link function table on Wikipedia, for example.