Welcome to the forum!
Are you asking someone to compare specifications for you? I am typing one-handed today, and using very weak wi-fi signal, so I am not going to do your searching for you.
If I recall correctly, the sensors are the same, or nearly so. If that is your only concern, there is little reason to spend more money.
The D7100 has a focusing motor, so it can be used with lenses, such as the AF-D lenses, that do not have their own focusing motors. Some of Nikon's finest lenses are AF-D, and do not have AF-S counterparts, such as my 180mm 2.8D and 16mm 2.8D. I do not think the D5300 has a focusing motor, so if auto-focus is desired, one must use AF-S lenses, which have in-lens motors.
Will the D5300 meter light with AI and AI-S lenses? Weak wi-fi means I cannot easily verify this for you today. I believe the D7100 can meter with AI-S lenses, as can its D7000 predecessor.
Does the D5300 have the two arrows inside the viewfinder, that help the shooter manually focus? I doubt the D5300 has this feature, whereas the D7000 does, so it is likely that the D7100 does, too. This is a "pro" feature, that was formerly found only in the top-tier professional bodies, first making its way into the semi-pro bodies in the D7000. Consumer cameras tend to just have a green dot to indicate focus, which is less precise than the arrows plus the dot. Some really fine lenses, such as the excellent new Zeiss Otus, are manual-focus-only, and some shooting is best done with manual focus, even with AF lenses.
Does the D7100 have more frames-per-second? That is important for some folks, who shoot active subjects, such as sports, birds, and wildlife.
The D7100 may well have a more substantial grip than the D5300. I shoot often in low light, and much of the time will have an LED or Speedlight in my left hand, so I like a camera with a substantial grip, to help me hold it without having to bend my thumb. Most consumer DSLRs have skimpier gripping areas. I handled a demonstrator D7100, and found it has a quite nice grip, better for my hands than the D7000 or D600.
I mostly researched the D7100 because I thought my wife might want to upgrade from her D7000, but she has indicated she is still content. It is possibe I will buy a D7100 so we can share batteries, chargers, Speedlights, and accessories while traveling, but I am more inclined toward a D800E, if I can afford one, some time next year.
OK, I will submit this when I can get a stronger wi-fi signal. When I started typing, no other member had yet replied.