The effect of dishwash detergent and clothes-washing detergent isn't too different, both are heavily caustic to remove greasy dirt although clothes powder has all kinds of additives like enzymes and TAED to enhance its effectiveness especially at low temperatures. The big difference will be that even an automatic powder is designed to foam a little for aesthetic reasons so sprayed at high pressure in a dishwasher it will foam like hell although the foam will be quick-breaking.
A hand-washing powder would probably fill your kitchen with foam!
The other difference is in the perfumes; dishwash powder perfumes are light volatile lemons etc. designed to do a short-term job of masking bad smells while clothes powder perfumes contain some interesting molecules like synthetic musks, ambers and sandalwoods, which are very cloth-substantive and won't get washed off or evaporated by hot air drying and ironing. These molecules are especially attracted to plastics so you might find that your plastic Ikea plates and cups come out smelling a bit tainted.
Added: don't put alloy bike parts in the dishwasher, the causticity of the powder actually erodes the surface of the alloy and dulls it. Don't ask how I know! It does work well on contaminated disc brake pads though.
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