There is a not lot of differences.Old Delhi existed long back during the Mughal period where as New Delhi was not there at that time.New Delhi was formed just about a hundred years back.starting with Edwin Luty endesigned area..with large bunglows and symmetrical roads in between.Old Delhi and New Delhi became common more in post independence India referring the older part of the city and the more planned newer part of the city.So to answer,they are same city,just different part called differently.Since they are not two different place.they are not at all far
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OLD DELHI- Old Delhi was the capital of ShahJahan.He found the city and named it as ShahJahanabad in 1639. The area was filled with beautiful houses and grand mansions. The main street of ShahJahanabad (or) Old Delhi was designed and built by Jahanara Begum Sahib-ShahJahan's lovable daughter.

The New Delhi shows modernity and progress. It was built by Sir Edwin Lutyens under the British.And the city is filled with wide streets,High buildings and is very much modernized.