There is no difference. "Turn a profit" is more idiomatic/figurative, as it includes the image of your fortunes turning, from loss to profit, in the near future, but otherwise the usage is much the same.
We're going to have to close the shop if we don't make/turn a profit again this quarter.
It even makes sense to use with an individual transaction
I don't know how but, even in this market, he still managed to turn/make a profit on the sale of that old house.
Note: Because "turn" is metaphorical, when using it imagine a literal change of direction, as in the similar idioms "turn a corner" and "turn over a new leaf", so it's kind of a decisive moment. Once a business "turns" a profit in a given time period (or transaction) it doesn't normally "turn" back.
Although, as with any metaphor, it can be used creatively:
We thought the business had turned a profit this quarter, only to have it run into a brick wall when we realized we hadn't set aside enough money to cover payroll taxes.