I would say a non-ideal diode (i.e., a real-world diode) would differ from an ideal diode in that it would show unit-to-unit variations in forward voltage and reverse current, as well as departures from the I/V characteristics predicted by theory. Most of these would be due to process variations and variations in the purity of the raw materials.
On the other hand, you could say a non-ideal diode differs from an idealized diode in that it has any forward voltage drop at all, and leaks any reverse current at all.
One really excellent resource for understanding the multifarious "non-ideal-ness" of all kinds of components is a book by Robert Pease, Troubleshooting Analog Circuits. Pease discusses all the quirks and foibles of resistors, capacitors, inductors, diodes, transistors, LEDs, opamps, comparators and sundry other ICs. The book is a bit pricey, but worth EVERY penny because it's simply chock-full of practical information.