How can I tell the difference between eczema and a normal rash?

Unclear. Even for medical experts, it is sometimes difficult to distinguish between different types of rashes in general. Sometimes, a skin biopsy needs to be done so that a piece of the skin can be studied under the microscope to determine the exact nature of a rash. See your doctor if you have a specific question regarding a rash.
Eczema is scaly. Eczema is chronic skin rash which is caused by an over-active immune system attacking and drying out the skin. I often consider it akin to "asthma of the skin" and, in, fact often goes hand in hand with asthma. While there is no such thing as a "normal rash" eczema is characterized by a dry, red, flakly scale which is often very itchy and often worse in cold dry weather..
A few clues. Eczema is the "itch that rashes"; rash doesn't itch it's not eczema. Itching but no rash? Not eczema. Eczema is chronic; doesn't go away although it moves around. Other chronic itchy rashes - hives & contact dermatitis. Hives are raised & bumpy; each bump lasts <24 hours. Contact derm is blistery, oozy & crusty. Eczema is dry & flaky. Look for eczema on elbow & knee creases, eyelids, neck, hands.
Atopic Dermatitis. "eczema" is a type of rash formally named atopic dermatitis. Atopic dermatitis is an inherited condition that's thought to be a histamine-driven problem, in much the same way asthma and hay fever are. The hallmark of atopic dermatitis is dry flaky skin, highlighting the importance of moisturizing. A combination of family history and the distribution of the eruption help make the diagnosis.