185/65/15 means that your tire's total height is 24.5 inches, it is 7.28 inches wide, and sits on a 15 rim.
205/65/15 means that your tire's total height is 25.5 inches, and it is 8.07 inches wide.
That means your tire will sit half an inch higher in the wheel well (i.e., you should gain about half an inch in ride height). Depending on the offset of your rims, your sidewall should move under half an inch outwards. That's as far as cosmetic appearances are concerned.
As far as your mileage is concerned, thicker tires will invariably mean more grip, slightly more rolling resistance, and thus, less fuel efficiency. It will probably mean better handling, softer ride, and better stopping power because of the larger contact patch. As an owner, that is something that you're the best judge of. Your own needs, and what is more important to you, will allow you to make the right decision. Best of luck.