Simply put: different class. One (D300S) is, albeit very old now, a pro/semi-pro body, the other (D3300) is a current (or almost current now that the D3400 is announced) entry level body.
Comparing D300S to D3300 is almost like comparing apples to oranges. I'll try to break it down for you.
What the D300S has over the D3300:
- Feature set (this covers all the camera can do):
- better AF system: 51 points vs 11 points
- faster shooting: 7 fps vs 5 fps (or even 8 fps with a battery grip attached)
- larger RAW image buffer: 20-45 vs 11 (for the D300S this depends on the setting, for the D3300 it's that since it doesn't have different settings for RAW)
- larger, brighter viewfinder: the one in the D300S has 100% coverage, D3300's - only 95%, meaning that in some cases with the D3300 what you think is out of the frame will end up in it; the pentaprism in the D300S is better at transmitting light than the pentamirror in the D3300 which means the D3300's viewfinder will be a little bit darker
- better handling
- more buttons and two command dials: most of the things directly related to shooting are within the reach of a button press or a button + dial
- more information at a glance (top info display and more things shown in the viewfinder display), hence you can more more easily change a setting without taking the camera down from your eye
- off-camera flash handling: the D300S is compatible with Nikon CLS (Creative Lighting System), meaning you can place a Nikon flash away from the camera, change some of its settings from the camera menu and fire it remotely using the built-in flash; the D3300 would be limited to what's commonly known as "dumb optical slave" - it can't fire a command signal before the exposure, meaning you will either have the camera's flash contributing to the exposure (not something you'll always want) or will have to block it with a thing like this (which blocks visible light, but leaves IR through)
- built-in AF motor, meaning the body will have AF with all Nikon and third party AF lenses; on the D3300 you're limited to lenses with AF-S designation (Nikon) and lenses with built-in AF motor (other brands), the rest of them remaining functional but MF only
- aperture sensing for non-CPU lenses - the D300S has an aperture "feeler" which lets you use lenses that do not couple electronically with the body: with the help of the feeler (and after setting things up in a menu), the D300S will know what aperture a lens is set at and will show you metering to help you with proper exposure; with such a lens, the D3300 will probably show f/0 and you'll have to guess the shutter speed
- longer battery life
- more robust body, better weather sealing, bigger dimensions (you might only consider this a plus if you're with large hands)
- ability to natively attach a battery grip - helping with shots in portrait orientation (also in landscape if you're with bigger hands), holds a second battery, reducing the need of changing batteries on the go (by "natively" I mean that the grip on the D300S communicates with the body via an interface connection that is on the bottom of the body and meant to do that; as for the D3300 and similar cameras - Nikon doesn't make grips for them; some third party brands do make grips for cameras with no native grip support, but they usually have to work around some things - i.e. no interface with the camera means they usually run a cable on the outside which attaches to the socket where you normally plug in a wired remote control; no control over which battery to use first if there's a place for more than one)
What the D3300 has over the D300S:
- newer, better sensor and image processor: more resolution and better image at high ISO settings
- better video capabilities: 1080p at 60 fps at most (D3300) vs. 720p 24 fps (D300S)
- lower weight
- (would only be a plus for people very new to photography) scene modes - I don't know your level, but with a bit of reading, you'll have gotten past these after two or three weeks of shooting; these are nothing more than combinations of exposure settings (and giving different weight to the different elements of it), AF settings, flash settings and picture controls (only valid if you're shooting JPEG; if you're okay with RAW, you have the freedom to do that afterwards); if you know what is important to each shooting scenario, P-A-S-M will be all you need, this is why most pro level bodies only have these
Of course these are only my observations. I might have missed something, so I might edit the post.