I have never read such a difference before, but i would surely take an effort to try to distinguish them on the way they are used.
Economical is a very wide word, it includes all economic activities. So if a mother is teaching her kids just for sake of saving some money, its an economical activity.
Commercial activity is related rather to profit oriented areas. Commerce, trade & commerce related activities.
Normal business activities come in Commercial activities.
So if a Business is Manufacturing cold creams it is an Economical activity. They are giving it for charity its still a economical activity and not commercial. When they sell it for No profit no loss its commercial activity. They sell it for profits, its a commercial activity. Till now we have not seen any Financial activity.
But if that business invest their profits in stock markets or banks or lend it, its a financial activity.
So financial activity is primarily connected with borrowing, lending. But over time definition of finance has increased which also includes Insurance, Mutual funds or I should use a wider word, Investing in Financial products.