Draft — every claim about the competitor below must be checked against their live site before publishing. Update “Verified on” when done.
The short version
Both plugins do the same core job: pack a whole site into one file and restore it anywhere. The free versions even share the same 512 MB restore limit. The difference is what happens when you outgrow it. All-in-One WP Migration sells the “Unlimited” extension on a yearly licence; Quick Restore Pro removes the cap for a single payment — and because Quick Restore reads .wpress files, you can restore every backup you already made.
When All-in-One WP Migration is the better choice
It has a longer track record, a very large install base, and a wide range of paid cloud extensions (Dropbox, S3, OneDrive and more). If you already pay for several of those and depend on one of the less common storage targets, switching may not be worth it.
When Quick Restore is the better choice
You want to pay once, you back up to Google Drive or the server, or you manage several client sites and want one 5-site licence instead of per-site yearly extensions. Migration is identical in practice: chunked upload, URL replacement and a confirmation before overwrite.