Color space is a model of how colors are defined. Some can define and use more color, or wider gamut. Color profile is a model to translate the colors to a particular destination, say a variety of paper and other devices. In other words, color space it relevant to the computer and the software in which you create your work, whereas color profile is relevant to where else it will be viewed be it on paper or other electronic devices.
That said, the confusion may come from using the same concept or model for both, particularly when outputting for other electronic devices. Printers and papers have their specific color profiles, say Epson Premium Luster on Epson 4880 printer. But when output for other electronic devices, you should consider converting your document to sRGB color space and depending on where it will be used, attach or do not attach the profile to the file itself.
I forgot to mention another sibling of these, Color Model. Color can be formed either through an additive process as in monitors using the primary colors RGB, or a subtractive process as in print using mainly CMYK colors. And, we have just scratched the surface of the color matter.