Once of the differences is the likelihoods for each model. In case readers can't remember, the likelihood encapsulates assumptions about the conditional distribution of the data. In the case of COVID-19, this would be the distribution of infections (or reported new cases, or deaths, etc) on the given day. Whatever we want the outcome to be, let's call it $y$. Thus, the conditional distribution (e.g. the number of new cases today) would be $y\vert t$ (think of this as $y$ conditioned on $t$).

* In the case of taking the log and then performing `lm`, this would mean that $\log(y)\vert t \sim \mathcal{N}(\mu(x), \sigma^2) $. Equivalently, that $y$ is lognormal given $t$. The reason we do linear regression on $\log(y)$ is because on the log scale, the conditional mean is independent of the variance, where as the mean of the log normal is also a function of the variance. So **Pro**: we know how to do linear regression, but **Con** This approach makes linear regression assumptions on the log scale which can always be assessed but might be hard to theoretically justify? Another con is that people do not realize that predicting on the log scale and then taking the exponential actually biases predictions by a factor if $\exp(\sigma^2/2)$ if I recall correctly. So when you make predictions from a log normal model, you need to account for this bias. 

* So far as I understand, `nls` assumes a Gaussian likelihood as well. Except now, we let the conditional mean of the outcome be non-linear. 

* The approach with `glm` frees is a little and allows us to control the conditional distribution as well as the form of the conditional mean through the link function. **Pro** These models are much more expressive, but I think the power comes from the ability to perform inference with a likelihood which is not normal. This lifts a lot of the restrictions, for example symmetric confidence intervals. The **Con** is that you need a little more theory to understand what is going on.

I've got to run, but this should be a good starting place. I'll add some details when I return.