Microdermabrasion will remove the top, dry, rough skin layers and will leave your skin softer and smoother. This procedure is similar to how your skin would feel after a nice facial. Microdermabrasion will have very little to no effect at all on acne scars (even mild ones), as these scars are within and often extend below the deeper skin layers. Even the deepest laser resurfacing, surgical dermabrasion, or strongest chemical peel will not completely remove or eliminate moderate or severe acne scars.
However, all of these options will have some beneficial effect on your skin, removing much of the sun-damaged irritations or rough patches (actinic keratoses), and improving, but not eliminating the acne scars.
As a general rule, you get what you pay for. Lighter chemical peels treat the most superficial layers, make minimal changes in true irregular scars or textural changes in skin, and heal quickly. That is why they are often done in a spa or skin care clinic by estheticians, and don't cost too much.
Stronger peels (higher concentration TCA or phenol peels) go deeper into the skin layers, remove more textural changes and scarring, but will still not be as aggressive as full-face ablative laser resurfacing, which gives the most improvement and best results, but costs more. No matter what procedure you have, the healing still takes about a week, and there may be redness that lasts somewhat longer.