Small non-coding RNA (sRNA) vs. micro RNA (miRNA)?
Small non-coding RNAs are NOT generally abbreviated as sRNA but as sncRNA, if anything. I say "if anything" because the "small" part is subjective and just a portmanteau descriptor. The straight answer to your question is as Forest has written: miRNAs are a specific type of small non-coding RNA, some of which appear to function in the regulation of gene expression — others remain to be characterized. Another problem with the 'small' epithet is that miRNAs may only be 20 nt in length, but are formed from pre-miRNAs of up to 1000 nt. 

Other categories of small non-coding RNAs include:

tRNA (transfer RNA) - originally abbreviated sRNA (for soluble RNA) - involved in protein biosynthesis.

5S rRNA - a component of the ribosome

7SL RNA - a component of the signal recognition particle involved in recognizing the signal peptide of secreted and membrane proteins.

snoRNA (small nucleolar RNA) many of which act as guide RNAs in ribosomal RNA modification.

snRNA (small nuclear RNA) many of which are involved in the splicesome, the ribonucleoprotein complex which removes introns from precursor mRNA (these have names like U1, U6 etc.)

SINE RNA: RNA transcribed from the short interspersed nuclear repeats (SINEs - Alu sequences in humans) which are thought to be 7SL signal recognition particle pseudogenes.

And, no doubt, there are some I (and science) have missed.