Applied a new technique on a facial recognition program and the new results are better than the previous one. Can I write a paper on this?
In general, yes, you can publish a paper. However, since you are basing your work on that of another you need to properly cite the earlier work. Check any copyright restrictions on the computer codes you use, also. It is possible that you will have to get permission if you use part of that code. But it is also possible that the authors already publish a license to use and extend. 

The only other issue is to find a journal or conference that will accept the paper. 

I would suggest that part of your paper discuss the earlier work in some detail and the results that it provides. All properly cited. Then you can discuss your new approach and how it improves on the original. 

It would also be good to explore alternate data sets to give yourself assurance that your result isn't too narrow or tailored to the specific data set of the original paper. You can even do a comparison on a range of data to get true confidence in the value of your work. 

I also think that the "new" approach would be more valuable if it is different conceptually than the earlier work, not just a more carefully coded, but essentially the same, algorithm. How is it new? How is it better?