Firstly, there are chemical differences between the two nucleic acids. DNA has the following structural properties:
-double stranded and anti-parallel polymer held together by (relatively) weak hydrogen bonds
-strong deoxyribose sugar backbone
-consists of the four nucleobases adenine, cytosine, guanine and thymine
-many C-H bonds that improve stability
-helical ('B' form)
On the other hand, RNA (mRNA, tRNA, rRNA, snRNA etc.) has a different structure:
-single stranded polymer that is more flexible
-ribose sugar backbone
-consists of the four nucleobases adenine, guanine, cytosine and URACIL (replacing thymine)
-C-OH groups make it less stable
-helical (but 'a' form)
The structure of the two polymers determines their functions. For example, the double stranded DNA is important for the replication of genetic information and conserving it for generations to come so the process can be repeated again and again (similar to photocopying a recipe book and then storing the photocopies in a safe for future use). This is known as semi-conservative replication. Each strand (after "unzipping" the DNA by breaking the hydrogen bonds) is used as a template to synthesise a new complementary strand. The strong backbone of deoxyribose sugar protects the base sequence, preventing random mutations changing the precious genetic code of a cell. This is reinforced by strong hydrocarbon bonds which increase its stability.
In comparison, RNA is single stranded meaning it can fold in on itself and form intrastrand complementary bonds (as we see in tRNA) and hence perform catalytic functions similar to the way a tertiary protein would. This catalytic ability plays an integral role in processes such as translation, where RNA is responsible for piecing together amino acids into a polypeptide chain (this occurs at the ribosome which is made of rRNA). Similarly, siRNA molecules in the cytoplasm utilise their catalytic ability to "chop" mRNA transcribed from a gene into tiny pieces. This prevents translation taking place and hence a protein is not manufactured, silencing the gene originally transcribed. Therefore, RNA has the ability to control gene expression. Finally, in contrast to DNA, the lack of stability means that RNA serves only as an intermediate rather than a "store" of genetic information.
In conclusion, structure and function go hand in hand. The differences in structure between DNA and RNA determines the differences in their biochemical functions in the cell.