ACE (AdWords Campaign Experiments) is a system that allows you to designate control and experiments within AdWords. While a control is usually your established ad, keyword, or ad group and your experiment is generally your challenger, Google doesn't force you to do anything except designate which is the control and challenger for auction purposes.
You then tell Google how often you want the control (or experiment) run in the auctions. For instance, a control could be run 60% of the time and the experiment 40%. Once this is designated, then when an auction (i.e. search) is run; Google uses your control a certain percentage of the time and your experiment the other times. Essentially, ACE is a complex ad serving system.
ACE does not tell you which is better (the control or the experiment). If you have a control ad group with 2 or more ads in it; then your campaign ad serving settings will dictate within the control or experiment auction, which ad will be served.
Adalysis allows you to analyze and take action from the data at the ad level. Adalysis allows you to:
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See winners and losers by many metrics
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Test ad templates across many ad groups
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Gives you insights into your overall testing
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Can help you forecast what will happen if you pause your losing ad
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Can automatically pause losers and push new ads live
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Can show you where you're not testing, or where you could do a better job testing
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And much more
In summary, ACE is an ad serving system. Adalysis is a complex ad testing system.