Correlation and regression do not distinct between "co-occurence" and "causality". These are different aspects of a a linear model. That's all. Correlation is a measure of the normalized covariance. This is very complicated to understand and presents some statistical properties of the data that are only of very minor impact for a technical/expert interpretation of the reaserch question (although is is frequently misused in this way). The regression uses this statistic internally to provide a more useful measure. The linear regression for instance gives a slope, that tells you how much the response changes when the value of the predictor changes. Based on the correlation, the regression analysis also provides standard errors or confidence intervals for these estimates.

I see the correlation as an quite uninteresing intermediate result. The more interesting thing is to know how much the values change (= the slope), and how precise this estimate is (= confidence interval). [Depending on the aim of the study it might sometimes be also interesting to provide a prediction interval.] Only then one can think if (and how much!) the protein concentration is determined by the RNA concentration, how intensively the RNA is translated or if there must be other processes playing a role in determining the amount of protein. So, if one really wants to learn something from the data, one needs a regression analysis anyway. Looking at correlation alone is an ancient habit that provides not much information, and the information it actually provides is very difficult to interpret scientifically.

In the recently finished MedStats class (StanfordX, 2014), the methods for correlated outcomes listed, depended on the outcome variable. It seems both, the given gene's expression (RNA production rate, internstity, etc.) and protein amount are continuous variables, hence, paired ttest (2 groups or timepoints), repeated-meaasures ANOVA (2+ groups/timepoints), or mixed models /GEE modeling (multivariate regression) are the plausible solutions.

If you consider some baseline measures (for both, RNA and protein), you should analyse the DIFFERENCE of given gene's RNA/protein and baseline; for this purposes, Wilcoxon rank-sum test can be used (you will have essentially 2 variables, the differences in either variable).

Correlation and regression do not distinct between "co-occurence" and "causality". These are different aspects of a a linear model. That's all. Correlation is a measure of the normalized covariance. This is very complicated to understand and presents some statistical properties of the data that are only of very minor impact for a technical/expert interpretation of the reaserch question (although is is frequently misused in this way). The regression uses this statistic internally to provide a more useful measure. The linear regression for instance gives a slope, that tells you how much the response changes when the value of the predictor changes. Based on the correlation, the regression analysis also provides standard errors or confidence intervals for these estimates.

I see the correlation as an quite uninteresing intermediate result. The more interesting thing is to know how much the values change (= the slope), and how precise this estimate is (= confidence interval). [Depending on the aim of the study it might sometimes be also interesting to provide a prediction interval.] Only then one can think if (and how much!) the protein concentration is determined by the RNA concentration, how intensively the RNA is translated or if there must be other processes playing a role in determining the amount of protein. So, if one really wants to learn something from the data, one needs a regression analysis anyway. Looking at correlation alone is an ancient habit that provides not much information, and the information it actually provides is very difficult to interpret scientifically.

I'd suspect most people use correlation because there are too many variables. Under a totally ideal, optimal unimpeded translation system you might expect protein and mRNA to share a clear relationship suitable for regression analysis, but this is very very rarely the case. mRNA transcription rate, secondary structure, subcellular location, stability, cellular balance of mRNP components, microRNA milieu, and availability of the translational apparatus are all factors that influence mRNA:protein relationship, and that's before we've even got to factors that influence protein level. It's not unheard of for a given signal to drive a large increase in mRNA level while simultaneously suppressing protein level. 

Yes, you can't make protein without mRNA, but the relationship between the two is usually very, very messy.

"I'd suspect most people use correlation because there are too many variables." - I also think so, but this is a "stupid" behaviour, IMHO. The slope and its confidence interval give you much more information, also that the relationship is (possibly) very messy (=huge confidence interval). It additionally tells you what protein levels one can reasonably expect (given the available data) for a given mRNA concentration.

It's true that there is a complicated relationship between mRNA and protein expression but we don't have to know all the fine details to predict important qualitative behavior.  That is, statisticians and biologists have different standards and purposes of knowing, sometimes.  

Please have a look at the following paper that examines expression relationships between RNA, protein and of genome-inserted fusion reporter constructs (which produces a modified protein) in context of somitogenesis, a directional oscillating process with a known periodicity, which allows one to predict spatially coordinated RNA/protein/reporter relationships (cf., Fig 2).  The relationships can be manipulated to accessible limits depending on how you affect protein stability or whether you keep introns in context of the reporter construct. 

Intronic delay is essential for oscillatory expression in the segmentation clock

Yoshiki Takashima, Toshiyuki Ohtsuka, Aitor Gonzáleza, Hitoshi Miyachia, and Ryoichiro Kageyama, PNAS 2010

Prediction would be great. However, as far as I know there is no program available which includes the amounts of mRNA, miRNA, mRNA-folding and (for E.coli) codon usage to name the most relevant factors.....

Correlation detects the amount of relationship between few quantitative normal variables. Its cant tell anything about the amount change of relationship if the values of variables changes. In regretion it is possible to tell if a predictor changes, what happends to response variable and this is the merit of regression analysis to correlation.

Do not forget about the post-translational events like the proteasome/ubiquitinization which also affect the relationship between mRNA and protein levels. I have seen increased mRNA expression with decreased protein level that was due to the activated protein degradation. It also depends on the stge of the coping mechanism. The relationship can change with time.

What you have read that correlation is the right analysis and not regression. See, no doubt, mRNA and protein synthesis are related, but then in what way regression anlysis is going to help you? 

the regression is used between independent variate and dependent variate so you can not use for genetic study such as transcriptomics ....