Alt should be used to specify alternative content. This should only display if the image is unavailable. Alt is how blind people know what an image is about.

Title may be used to add additional information, which browsers may render as a tooltip.

You should always specify alt. If the image is purely decorative, use alt="" (blank). You may also specify title. Title is optional; good link text usually makes title redundant.

Search engines will not penalise you for using these together; but don't use them for keyword stuffing -- that will likely carry a penalty.