Don't Starve Together - Alone mode vs Don't Starve

Well...

Don't Starve has a couple small unique things, since they couldn't be reworked for multiplayer. Don't Starve Together has significantly more stuff.

Don's Starve has a few exclusive mods.

Don't Starve Together has skins, but some mechanics that were reworked require more players -- All bosses have 2x the HP they had in Don't Starve, except for the Dragonfly which has 10x the amount of health it normally does.

All-in-all, I recommend getting both! Don't Starve has Shipwrecked and Don't Starve Together is just awesome!

Based on you comments I would recommend playing Don't Starve Together Alone, first.

DST has unique content and game mechanics not available in DS. If your trying to warm up to play with a friend then best to warm up with all the same mechanics.

You will find that playing DST alone is more challenging then if you were to play DS. However, you will find that if you play DS then try to transition that knowladge to DST, there will be areas that just seem "off" because of mechanic differences.

A great example is looking at the map. In DS, when you look at the map the game pauses. You can spend a long time looking at the map to try to figure out the place you want to go "today". In DST the game does not pause to look at the map. If you spend too much time trying to figure out where to go you will run out of day time. This small change can be a hard habit to break when transitioning to DST.