Half-LIfe: Source is a, ehem, "remake" of Half-Life 1, which in this case means "we took HL1 recources and transfered them off GoldSrc engine to Source Engine". And yes, it contains single-player campaign of HL1... DeathMatch portion was released serapately later. Though now that you can only purchase bundle of Source with DM, it doesn't matter much. Yes, it comes with HD models from Blue Shift bundled
So, what remake brought? Nothing major. Just small things here and there. First, default things from engine changes - dynamic shadows from flashlight that HL2 Episode 2 brought into engine, for example. There is more of metallic reflective textures, that look ugly in the darkness. The water was changed into HL2 style one... Means more transperency yet the rippling animation was removed. Enemies don't have death animations anymore - they use ragdoll... Which sometimes looks clunky with low-polygon monsters. There are even occasional decoration object that now uses physics, like hooked boxes. And you don't throw boxes away over the hills when you try to move box sideways. At least it all comes along with HD Models pack from Blue Shift.
One of the non-graphics changes got to be improved AI, which I also think came. NPCs always react finely, while in original they would often first turn around slowly to face the threat. Security officers that follow you don't get lost as easily, you can even forget about babysitting them. And where in original they had "shoot at some aliens" as high priority, "follow the player" seems to be higher one in this remake... Which means that they will not shoot a thing if they will be busy following strafing player. Guards are better at guard duty, yep. On one of the notes, flying enemy with huge brain in final parts of games is now more aggresive, attempting to get into your face, all the time!
Some of the damage values were changed. The biggest example got to be clowbar, it got buffed... Now, just like in HL2, it takes only one hit of this tool to bring down Headcrabs and Barnacles.
Some of small bugs in HL1 were fixed, some new small bugs were added. Oh yea, and music stops when you go to the next area.
But overall, original is safer bet as mods for HL1 will work with original, not Source remake. But Source one isn't bad either. Much better looking today, at 2014 year thanks to engine updates, while at time of release it didn't look like it added much.
Deathmatch multiplayer part of remake was released after single-player part and comes by default with bundle on Steam... But let's just say, it's dead and not worth checking out.