There is clearly difference.
初め = first
始め = beginning, start, launch
Difference between those two is "FIRST" and "BEGINNING or START". Hence, "how do you do?" goes to the former but not the latter one. Now you know what comes next?
When you want to greet someone you've FIRST met, you will say
初めまして = nice to me you, how do you do?
When you will START or BEGIN a new business, you will say
あたらしい しごとを 始めます。