Extra virgin olive oil and expeller pressed grapeseed oil are both healthy cooking oils that contain good fats. Decide which to use based on the flavor you prefer and how well the taste of the particular oil balances with other ingredients you include in a dish. Also consider the cooking techniques you'll be incorporating.
About Olive Oil
According to Mort Rosenblum's "Olives:The Life and Lore of a Noble Fruit", olive oil was already a staple in Mediterranean regions during biblical times. Since the 1960's it has grown increasingly popular among Western cooking enthusiasts and dietary experts. Extra virgin oil is extracted by crushing the fruit of the olive tree, then separating the oil from the pulp and pits, usually with a centrifuge. Subsequent pressings yield inferior grades of oil, such as pomace. Olive oil contains monosaturated fat, which lowers levels of LDL, or "bad" cholesterol. (healingdaily.com link in references)
About Grape Seed Oil
Expeller pressed grapeseed oil is produced by crushing grape seeds, extracting the oil without using toxic chemicals. Like olive oil, grapeseed oil has been linked to cardiovascular health, improving cholesterol levels by changing the ratio of HDL (good) cholesterol to LDL (bad) cholesterol. It is also high in antioxidants, which can counteract the damage done by carcinogenic chemicals in our daily environments. It is a common ingredient in skin care products as well. (knowledgrush.com link in references)
Cooking With Olive Oil
Olive oil has a rich and complex flavor, which is created by the variety of olives it contains, the care with which they have been handled throughout the production process, and the climate and growing conditions shaping each particular vintage. It is perfect for cold salads, where its taste can truly shine, but it is less appropriate for cooking, especially deep frying, because it has a low smoke point. This causes it to break down and take on an unpleasant flavor at a relatively low temperature.
Cooking with Grape Seed Oil
Unlike olive oil, grapeseed oil has a milder, more neutral flavor. This makes it an excellent choice for baking, especially with sweet ingredients, and for recipes where the strong taste of olive oil is inappropriate. Grapeseed oil has a relatively high smoke point so it is ideal for stir frying, a technique which cooks food quickly on a high flame. It is a good oil for deep frying, and it can be used multiple times because it doesn't break down easily.
Expeller pressed grapeseed oil tends to be less expensive than extra virgin olive oil, especially the imported, artisan varieties. It can also have a smaller carbon footprint. Local food aficionados who do not live in olive producing regions can often find grapeseed oil that has been produced closer to home, and the fact that it is produced as a byproduct of another industry--wine--means that agricultural land is not usually dedicated to its production. Olive oil, however, can offer nuances of flavor that are not usually found in grapeseed oil.