I wrote a simple VB.NET application which works fine on the development machine.
However on another machine, even though the .NET Framework is installed, it crashes
Apparently, it seems to be a problem with the "PowerPacks"; these are additional controls to the ones given by default in Visual Studio (in the form design part, under "Toolbox"). If you install the required package on the target computer, everything should be fine.
I had this problem in the past and my recommendation is not relying on these elements: they include nice-to-have features (in my case, it was an elliptic shape) but might be replaced with a bit of work and simpler objects. If you want to sell your program to a more or less wide audience, relying on this might become a problem: this package is not installed by default on virtually any computer and thus you would have to ask your clients to go ahead with the installation. Locating these specific controls is straightforward: on the "Toolbox" they should be listed under "PowerPacks" or similar.
After reading the Hans Passant's comment, I do recall that a .dll file has to be put in the same directory than the executable, but I am under the impression that the PowerPacks package from the provided link has to be installed anyway (at least, on computers not having Visual Studio).