You accomplish your goal with SyncBackFree (portable version available here), SyncBackSE, and SyncBackPro.
You can compare the different editions here. SyncBackFree is gratis, SyncBackSE is within your price range, and SyncBackPro is at the high end of your price range.
They are all from 2BrightSparks. I have used their software for many years and find it to be of high quality.
The first step will be to use any tool you like (such as
attrib, which is included with Windows) to enable the
Archive bit on every file on your drive partition(s).
The next step will be to create a profile in any of the SyncBack products to only copy files with the
Archive attribute enabled, and to remove the attribute once the copy operation is done.
To accomplish this, create a new SyncBack profile and go to the
Compare Options tab in the profile's setup window (the tabs are positioned vertically along the left edge of the profile's setup window). In that tab, check the checkbox for "Only copy files that do have the archive attribute set".
Next, go to the profile's
Copy/Delete tab, and then the
Advanced sub-tab (the sub-tabs are along the top of the profile's setup window). Now, enable the "Reset the archive file attribute on files once they have been copied" option for the profile.
Outside of these somewhat hard-to-find settings, everything else should be relatively obvious. For each profile, you can specify which folders and files to backup. You can create multiple profiles to break up your backup tasks into smaller segments, if you like. This can be very nice when you are actively working on a project and want to make sure all of the project's files are fully backed up.