Originally Posted by milkbaby

...The more relaxed geometry will usually have a longer wheelbase which tends to make the bike more stable but less twitchy and react slower to steering inputs.

The head tube will usually be taller, so this will affect how low you can position the handlebars and how low you can get for aggressive aerodynamic positioning.

The top tubes tend to be shorter and the head tube angle slacker, so these two factors tend to put the rider into a more upright position which many people find more comfortable than the prototypical long and low position of many aggressive race bikes. That also means your center of gravity will be higher, so it may be harder to aggressively carve corners and turns. The slack head tube angle also increases the trail which makes the bike more stable but with the same trade off of giving up quicker steering response.

Some of the "endurance" relaxed geo bikes also have additional features which are aimed to increase vertical compliance for greater comfort.