If you use this water to convert mRNA to cDNA, it must be RNase free water so as not to cause the RNA breakdown of your samples. If you use it in the RT-PCR mix, it must be DNase free water in order not to breakdown the cDNA.

Many RNases are tolerant to denaturation and will refold to restore activity even after boiling. So just because water is sterile does not mean it is free of RNase. Hence the use of specially treated water to ensure that all RNases have been properly inactivated. DNases are less of a problem but again sterility does not equal no contaminating enzymes, depending upon your mode of sterilization.

Actually, it is not so obligatory in all cases. In my practise I have always used water of two step purification: distillation then de-ionization (like Milli-Q), and have never faced DNA/RNA degradation (apart from cases when a working place itself was contaminated). Enzymes cannot go through distillation along with water, so as long as you keep your water equipment and vessels clean, your water will be suitable for most applications like PCR.

purified or sterile water can have a high pH and minerals that can interfere with reactions that require specific salt and pH conditions. But with the RNase free dH2O, it's prevents contamination and helps in preventing breakdown of your samples.

Hello,

double sterilized distilled water after autoclave also can be a perfect choice, due to deactivation of all types of enzymes. It depends on purification filter which you have used. If you are not sure about the filtered water, it is better just use the commercial prepared RNase and DNase free distilled water.

Sterile water may still contain nuclease enzymes which will interfere in pcr either by degrading the DNA sample or giving false positive results