The fundamental difference is all-in-one vs. separate CPU and display. All-in-one is more compact, somewhat neater, and offers greater simplicity - slightly fewer connections to make, fewer decisions (no need to shop for a separate display).
Comparing specs should be easy. As has been noted, the iMacs are more up-to-date and have superior specifications.
If you're interested in the lowest-priced entry into Mac computing, get a Mini and connect it to a display you already own. However, if your need for computing power grows and you've grown to love Mac... you may end up buying an iMac or MacBook in a couple of years anyway.
In theory, all-in-ones can more economical (single enclosure, single power supply), but you can harness an existing display to a new or buy a bargain display and also save. If you're looking for a higher quality display, the 4K and Retina iMacs have actually been a "value proposition."
From a standpoint of environmental waste, if you have a perfectly serviceable display, why replace it prematurely? If a display replacement is imminent, then an iMac may be less wasteful than buying separates.
If you're interested in a high-end display, you'd likely want a CPU with performance to match (better graphics chip in particular). Since today's Minis are outdated and underpowered in that regard, you're better off getting a 4K or Retina iMac with graphics capabilities that match their excellent displays.
Then there's the personal vanity side of things; having a sleek iMac with it's black Apple logo on exhibit vs. an inconspicuous Mini harnessed to a big display from another company.
The era of separate CPU/monitor configurations is drawing to a close. Laptops are all-in-ones, smart phones and tablets are all-in-ones, and desktop all-in-ones continue to gain sales over separates. While there will always be those with compelling reasons to mix-and-match components, the vast majority of computer buyers no longer need to do that.
Hardware tinkerers hate all-in-ones - it's very difficult, sometimes impossible, to replace/upgrade components. But, like auto repair, far fewer people do their own computer repairs today. Ease-of-repair comes at a cost - additional components, fasteners and connectors that actually reduce long-term reliability... If you haven't been performing "surgery" on your PCs, there's no need to start doing it on a Mac.
