RMA is the Robust Multichip Average. It consists of three steps: a background adjustment, quantile normalization (see the Bolstad et al reference) and finally summarization.
Hypothetically you will get the same results from both places, provided you have consistent settings in affy/BioConductor and RMAExpress. Some people prefer the power and flexibility of R and others like the point and click simplicity of a GUI. RMAExpress caters to the second option. Since RMAExpress outputs the computed expression values to a text file, you may of course load the expression measures into R and use features of Bioconductor for the analysis of your gene expression values. You can of course open the results file in any other application that supports importing plain text files
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