Let's say I had contracted with a third party to design my beautiful company logo.
Now two years later I need a version without the fake glassy front. If I have the original, this takes the time it takes for me to launch PhotoShop Elements, uncheck a box next to the "glassy button" layer, and export the result to web.
If I only have the PNG file, I'm basically out of luck if I can't get ahold of the artist and will have to have someone else redraw the logo from scratch.
And not only should the PSD files be in your possession, they should be in your source control system, along with all the other critical changeable items you've spent so much time and money creating.
I've been in a situation where I have literally 100 toolbar icons in PNG form and I have no easy way to make a bigger size because the outsource artist has consistently neglected to deliver the original files. Frustrating.
Also, once you have the PSD files, have a look and see if they are well organized with ample use of layers, and layer naming. Maybe, there is no way to turn off the glass button by hitting the hide layer checkbox, maybe there is only one layer, and the artist is incompetent. Maybe that's why you had to ask.