What is the difference between Curve Fitting and Regression(Machine Learning)?

I know that Machine Learning regression algorithms try to find the function of the data. That is, if we have 1000 data points (x,y), to find a general continuous function that follows the trends of the data and which can provide estimated values y for other x that we do not know their actual value y.

But curve fitting doesn't do very similar thing? I mean, yes, we may have a function this time and not only those 1000 points, but it also tries to find a general continuous function that follows the trends of the data.

Therefore what is the most actual difference between the two? The algorithms that work for Regression in ML can't they be applied at curve fitting as well?

Update 28 Apr 14

I am reading this paper and it says that it uses Curve fitting and Regression techniques. What is actually their difference? What algorithms/techniques can be considered as Curve Fitting and what as Regression?

Yes, curve fitting and "machine learning" regression both involving approximating data with functions. Various algorithms of "machine learning" could be applied to curve fitting, but in most cases these do not have the efficiency and accuracy of more general curve fitting algorithms, finding a choice of parameters for a mathematical model which gives "best fit" (variously defined) to a data set.

In curve fitting we are often interested in parameters for a mathematical model based on a theory of cause and effect underlying the data, which may include random or systematic errors.

An attraction of "machine learning" is to give machines a task of "discovering" information through data mining. E.g. machine learning algorithms might be applied to optical character recognition.