siRNA is something that you buy on the market, created by the biotech Companies, miRNA is something created by the nature.. Joking :) The most relevant difference is that one siRNA targets only one mRNA, one miRNA can target up to thousands.
Although previously answered years ago by sufficient individuals (Links below).
My own studied explanation is miRNA & siRNA have these differences, 1) miRNA are from a single strand and folded into double strand where they are processed by Dicer, each of which can lead to differing lengths of mature double-stranded miRNA without un-necessary overhangs; while siRNA are segments of RNA, initially single-stranded (so viral or alternative splicing by-products from organism), which is turned into double-stranded small RNA by polymerase, and followed by Dicer activity. 2) miRNA are less specific than the siRNA with the RISC machinery, where miRNA pairing is attributed to seed-region pairing and siRNA are expected for perfect complimentary with target mRNA.
As Anastassia mentioned, siRNA are almost more likely to be used for gene silencing function experiments as they have significantly less off-targets. miRNA are less specific, as mentioned earlier.
I know I was confused by the literature initially, but know that siRNA are although preferred in plants, but miRNA are also present especially during development where they are more useful than siRNA. Vice-versa, animals prefer miRNA, but also have siRNA.
I hope this will help you understand small RNA a little better. For a more detailed explanation, I would recommend the attached publication to read.