Both words mean "a diffuse inflammation of the skin and subcutaneous tissue due to infection", usually with Gram-positive bacteria. Most scholarly sources now use the two interchangeably, as a quick search of PubMed will show. "Phlegmon" is a rather outdated term, originally used as a counterpoint to "abscess".

Here is the definition of cellulitis from Bolognia's Dermatology, a standard textbook: "Cellulitis is an infection of the deep dermis and subcutaneous tissue caused most commonly by Str. pyogenes and S. aureus".

A distinction between "cellulitis" and "phlegmon" in a dermatology setting did originally exist, with "phlegmon" sometimes considered the outcome of cellulitis, but this distinction has been lost at least since the 1900s, as suggested in this 1935 article (sadly in Portuguese!).
http://www.rborl.org.br/conteudo/acervo/print_acervo.asp?id=...

There may still be a distinction in other contexts, but if we're talking about soft tissue infection, I see no problem in using both interchangeably, although I see absolutely no reason to translate "cellulitis" as "phlegmon" when the former is much more widely used and understood in English!