The short answer
Unscented means a product is meant to smell like nothing, and that can be done by adding ingredients that cover up the smell of everything else. Fragrance-free usually means no fragrance ingredients were added at all. Neither word has a strict legal definition for cosmetics in the US, so the ingredient list is the only place to get a real answer.
Why it happens
Many raw ingredients carry a smell of their own. Butters, oils, plant extracts and some preservatives all have one. A maker who wants a product that smells like nothing has two ways to get there. One is to choose ingredients with very little smell and add no scent. The other is to add a small amount of a scent ingredient that quietly masks the rest. Both products can end up with the word unscented on the front.
Because the word is not tightly defined, a maker is free to use it on either kind. Two jars with the same label can be quite different inside. Fragrance-free is a little more specific in everyday use, but it is not tightly defined either, and a front label is marketing, not a promise you can check.
The ingredient list is where the facts are. You will often see the single word fragrance, or parfum. That one word can stand for a blend of many scent ingredients, and the maker is not always required to name each one. For someone who reacts to scent, that is a hard thing to trace. If a reaction shows up and the list only says fragrance, there is nothing to point at.
Essential oils are scent ingredients too. They are sometimes assumed to be a gentler choice because they come from plants, but a plant source only tells you where something came from. It does not tell you whether your skin will like it. Citrus oils and lavender oil are popular and can still bother skin that runs sensitive.
Not everyone reacts to scent. Many people use scented products for years and never notice a thing. But for those who do, the signs are usually redness, a stinging or burning feeling soon after application, or an itch that shows up a day later. Sensitive skin tends to notice small things, and scent is a common one.
What helps
Turn the jar over and read the back before you read the front. Look for the words fragrance and parfum. Look for plant oils, which on a label often appear by their Latin names followed by the word oil in brackets or after the name. Look for names that end in ool or ene, such as linalool or limonene, which are scent components that sometimes appear on a list.
Scent ingredients are usually in small amounts, so they tend to sit near the end of the list. That is a useful habit: read from the bottom up. A short list is easier to read. The fewer ingredients a jar has, the fewer unknowns you are carrying.
Keep a note on your phone of anything you have reacted to. When you can say that a certain oil or ingredient bothered you, label reading gets much easier, and your pharmacist can compare products against that note with you. Bring the jar to the counter and ask. Reading labels is a normal part of a pharmacist's day.
Patch test anything new, even if the front says fragrance-free. Put a small dab on the inside of your forearm, leave it uncovered, and check it each day in good light. If the skin stays calm, go ahead. If it turns red or stings, wash it off and stop.
Remember that scent reaches skin from places other than your moisturizer. Laundry detergent, soap, a partner's cologne and fabric softener can all leave scent on skin and clothing. If your skin is touchy and you have changed your skincare but still have trouble, it is worth looking at those too. A week or two with plain, unscented laundry and soap is an easy experiment, and it costs very little.
Finally, give a new product a real chance before you judge it. Skin that has just come through a dry spell or a change in weather can sting for reasons that have nothing to do with the jar. If you can, test in a calm stretch, and change one thing at a time.
When to ask for help
Ask your pharmacist or your doctor if redness or stinging keeps returning even after you have stopped new products. Ask if a rash spreads, blisters, or weeps. Ask if you cannot tell which product is the cause and you want help narrowing it down. They can read a list with you and tell you what to look at first.
Some reactions are an emergency. If your lips, eyes, tongue or face start to swell, or if you have trouble breathing, call 911 right away. Do not wait to see whether it passes.
The AHOTO note
By the standards above, Golden and Glowabe are neither fragrance-free nor unscented. Both contain a light essential oil blend for scent. Both also contain oats and soy, and we recommend a patch test first. Pure Shea has one ingredient, cold pressed unrefined East African shea butter, and nothing added. It does have the mild, earthy smell of the butter itself, which is the butter and not an added scent.