This is not a comprehensive answer but, in a nutshell, I could say the following about each:
Limma (as it was/is used for microarrays):
After background correction, quantile normalisation, and log base 2 transformation, limma fits a linear model to your data and then performs statistical comparisons via linear regression
DESeq for RNA-seq
DESeq2 performs a 'geometric normalisation' and then comparisons are generally conducted using the Wald test. DESeq2 also allows you to transform your normalised counts using a regularised log transformation, which deals with transcripts of low counts, or a variance-stabilised transformation, which deals with the high variability of low/high counts that you get from RNA-seq data.