The only difference between crimini mushrooms, white button, and portobello is their age. The crimini is really just a more mature (older) version of the common button mushroom. All of them come from the same fungus, Agaricus bisporus. However that doesn't mean their nutrition is the same. Just as an unripened vegetable can have different nutrition than a ripe one, the same holds true for funghi.
Do mushrooms have nutritional value though? That depends on your definition. If you are talking about protein, carbs, and the vitamins and minerals found on a standard nutritional label, then the answer would be no. However if you're talking about phytonutrients - things which are not covered by the label - then the answer is a resounding yes.
There a number of phytonutrients in crimini which have been suggested in research as possibly having a positive impact on our immune system. Those include beta-D-glucans, p-tolyl-hydrazine, APO (2-amino-3H-phenoxazin-3-one), and fucogalactans (1). Possible anti-inflammatory health benefits are another which have been studied, albeit many of those studies have involved animals rather than humans. Their findings have suggested that compounds found in crimini (as well as other mushrooms) may help block the production of pro-inflammatory molecules (2).
USDA Database for the Oxygen Radical Absorbance Capacity (ORAC) of Selected Foods, Release 2 - Prepared by Nutrient Data Laboratory, Beltsville Human Nutrition Research Center (BHNRC), Agricultural Research Service (ARS), U.S. Department of Agriculture (USDA) - May 2010