I view the difference between these as the difference in extraction and brewing style. Aeropress is ground finer and extracts faster. French press is ground coarse and extracts slower. Drip/Chemex is in the middle. The extraction will impact the flavor, mouth feel, and overall characteristics. Also, all these, but French press, use filters so oils will be partially or totally removed. French press doesn't use paper filters and, therefore, allow for oils to remain, adding a different complexity.
French press can be "muddy" in a lot going on in the cup due to extraction and brewing style (immersion). Aeropress is partially similar in brewing style (starts as immersion), but is filtered and is cleaner. It can make somewhat of a pseudo espresso. Drip, Chemex in particular, I find to be the cleanest and brightest in flavor.
It's fun to play around with them or find a coffee shop that offers these various methods so you can taste the same coffee bean made different ways. The taste variation is definitely noticeable.