Difference between confidence intervals using standard error and standard deviation.

Can someone please explain the difference between an interval built by:

x̅ $\pm$ (Critical Value) X (Standard Error)

and

x̅ $\pm$ (Critical Value) X (Standard Deviation)

It's not clear to me when to use which. Thank you.

There are many kinds of confidence intervals for a variety of population parameters. My guess is that the one described here is what you have in mind:

If $X_1, X_2, \dots, X_n$ is a random sample from a normal distribution with mean $\mu$ and standard deviation $\sigma$ (both parameters unknown), then a 95% confidence interval for $\mu$ is of the form $$\bar X \pm t^*S/\sqrt{n},$$ where $\bar X$ is the sample mean $S$ is the sample standard deviation, and $t^*$ cuts 2.5% of the area from the upper tail of Student's t distribution with $\nu = n-1$ degrees of freedom. The standard error is $SD(\bar X) = \sigma/\sqrt{n},$ estimated as $S/\sqrt{n}.$

Notes:

(1) In the situation above, if $\sigma$ is known and $\mu$ is unknown, then a 95% confidence interval (CI) for $\mu$ is of the form $\bar X \pm 1.96\sigma/\sqrt{n}.$ Because $\sigma$ is known, it is not necessary to estimate the standard error $\sigma/\sqrt{n}$ and one may use the standard normal distribution (instead of a t distribution) to get the value 1.96.

(2) There are CIs in which the standard error happens to be the same as the standard deviation. (One example is a CI for Poisson mean $\lambda.$ However, I doubt that is what you have in mind.)

(3) A 95% confidence interval for normal SD $\sigma$ is based on the chi-squared distribution. Because that distribution is not symmetrical, values cutting 2.5% of the probability from each tail (separately) are used, and the CI does is not of the form (point estimate) $\pm$ (standard error).