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

The two versions of the game are balanced differently in terms of difficulty.

As you can read in this discussion on the Steam Community, one of the devs says exactly that: you can play Don't Starve Together as a solo game, to start to grasp the mechanics, but as you start to progress you'll find harder content than the "base" version. Some of the challenges that Together makes you face are not meant to be tackled by yourself. Take for example the Ewecus: its spit can trap you for a few seconds and, while it could be overcome by having a friendly Pigman, having another player come to your aid makes the fight significantly easier.

In addition, at the moment of this writing, Together features the content of the base game and Reign of Giants, but not the Shipwrecked content.

Many things were modified or rebalanced too, due to the intrinsic power of some of the characters, and special encounters and events have been added to adapt to the multiplayer aspect. You can read the whole list of differences on the wiki page.

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.