The Change History in the AdWords UI lets you see the changes that were made in an account (e.g. budget adjustments, keyword edits, and changes in campaigns and adGroups). This
article explains it in detail.
In the AdWords API, what seems to be close to this feature is the
CustomerSyncService but it only provides what objects changed during a time period, and not as specific as change in budget that you specified. You may keep an eye in our
blog to check on future announcements regarding this feature.