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Time series regression function looks very similar to the "regular" linear regression. But they're different. TS regression has auto correlation issues, but regular linear regression assume no auto correlation. My question is, what's the relationship between these two regressions? What's the similarity and the difference?

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Your question contains much of the answer! Time series regression is the study of autocorrelations (correlations between the "past" and the "future" of your time series).. Linear regression assumes all your observations are independent from each other.

In other words, let's say you have a sample with the following target values:

A: $3,1,3,1,3,1,3,1$

B: $1,1,1,1,3,3,3,3$

Linear regression will treat both datasets as the same. However, if you fit a time series model instead, the results will be way different.

Also, in linear regression we study how other predictors ("regressors") influence our target value. While this can be the case in time series analysis ("external regressors") it is possible fo study a time series working only on the values of the series itself

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