I use both but I definitely use embossing folders more often than Texturz plates - and I own lots of folders but just a few Texturz plates. With a folder, the CS gets sandwiched between two sides and the pressure means you get a really deep impression. I love the fact you can ink an embossing folder before embossing. I recently found a bisque pottery technique which I absolutely love. Here's a link to the blog post and Di also has a video showing how to do the technique.
Stampin' With Di: Bisque Pottery Technique
You could probably ink Texturz plates but I never have and all my plates are black and I should think it's not going to be as easy as with a clear embossing folder.
With the Texturz plates, they're only one sided so the impression is not as deep because there's no sandwich so the pressure is different. What I like about the Texturz plates is that they're a larger size than the folders so if I'm making a square card and want some texture I'll reach for a Texturz plate.
I have a Big Shot and I experimented with the Texturz plates until I got results I was happy with. I'd recommend trying and experimenting with your own machine. I add one piece of CS as a shim and I also spritz the CS I want to emboss with water before embossing with the Texturz plates and that seems to help get a deeper impression.