I'll twist things around a bit.
Your preferred uses (children, products, and nature -- so long as you aren't referring to "action wildlife photography" when you say "nature") don't indicate a need for a high-performance camera (you do not need a camera that can shoot 14 frames per second with a staggeringly sophisticated focus and tracking system.) This means you have a lot of options (a good thing too because those cameras that shoot 14 frames per second with those amazing focus systems are all well over the $3k limit.)
Fundamentally, if we ignore the notion that your Canon was a film camera and your Nikon is a digital camera... one other very significant difference is the image size. The Canon film camera had a 35mm negative frame (24mm x 36mm) The Nikon has an APS-C size sensor.... so named because it's "roughly" the same size as a single frame of APS-C film (Advanced Photo System - Classic) and that had a 50% crop factor (by area... not dimensions... in other words it's not half of the 24mm x 36mm frame size... it's closer to 2/3rds or around 15 x 23mm.)
Anyway this smaller sensor means the reflex mirror is smaller, the pentaprism is smaller, the viewfinder is smaller, etc. But it also means the "angle of view" is tighter. That reduced angle of view may mean you'll chose a lower focal length lens to take the same shot or it may mean you'll choose to shoot your subject from a different distance so that, in the end, you get "roughly" the same framing that you had planned to compose.
When you make those behavioral adjustments to compensate for the crop-factor difference, you will end up changing things like the depth-of-field.
You may want to go look at and evaluate a "full frame" camera vs. a "crop frame" camera.
In Canon, the 6D, 5D III, and the 1D X are all "full frame" bodies -- with sensors as large as a 35mm film frame. Every other Canon DSLR uses an APS-C size crop-frame. The 1D X and 5D III are above your $3k budget... which leaves the 6D within budget.
In Nikon, the full-frame bodies are the D610, Df, D800 series (800, 800e, and 810), D4, D4s, and D3X and everything else is APS-C crop-frame. But most of those (D3X, D4, D4s, 810, 800E) are above your budget. The D800 (the oldest of the D800 series) is $2999.95... without a lens. So we may as well add that to the "over budget" category. The Df is an interesting camera ... it's about $2750 (without a lens) and this camera has a 50mm f/1.8 "kit lens" bringing it to $3k (most cameras will have a standard zoom... meaning a little wide-angle and a little telephoto in the range but nothing too extreme.) Also the Df "looks" like an older camera -- it has a nostalgic looking body that will probably remind you more of your Canon film camera (even though it is an advanced digital camera.) That means that like the Canon... the single camera body that probably stands out in terms of affordability is likely the D610.
The Canon 6D and the Nikon D610 are both considered "entry level" for full frame cameras. Previous to the Canon 6D and Nikon D600 (the D600 was discontinued and replaced with the D610) no full-frame DSLR had a price tag less than $2k -- most being much closer to $3k at the low-end and going well-up from there. Full-frame DSLRs were strictly the kit of the professional ... or the very enthusiastic amateur with the financial means to afford them. However... "entry level" for a full-frame camera is still a VERY good camera.
You would likely notice the larger and brighter viewfinder. You may be more comfortable with the noticeably larger size camera body.
But the really big difference will be the return to focal length and DoF behaviors that you were used to when you used a 35mm film camera.
One other thing... but this will be true of most any DSLR camera currently marketed... Your D80 was old and a bit limited in terms of sensor performance. Everything currently marketed today will have significantly better sensor before ... pretty much in every way (ISO range, less "noise" at high ISO, better dynamic range, better bit depth and more adjustment latitude when shooting RAW, etc. etc.) If the quality of the images you were getting with the D80 were disappointing then you'll probably be impressed with all the current offerings (regardless of who makes it.)
Given your long experience with 35mm film, your background, education, your budget, and the notion that you want something not necessarily "pro" but it sounds like you want something "near pro" ... I think you should probably give those full-frame bodies a look.
. Mainly I take photos of children, and products, but also nature. 
