I want to know the pros and cons of each. I am running a small group of three players and I want to see which would be preferable.
I think it comes down to personal preference. A home campaign is good because you have freedom as a DM. I don't run APs for my game, although I'm missing out on the goodness that comes with them.
The APs come with extras you can use later in the back. Easter eggs, if you will.
Building a game from scratch can be really satisfying, but it does take a lot of time. Whether or not the workload fits into your life is something that I can't answer.
On the other hand, with a small group of three, you would want to do some encounter rebalancing, if you were going to use an AP, unless you send an NPC along with the party to bolster their capabilities. Some of the things are written in those to be quite dangerous for a 4 player party, already.
What I will say is that published adventures like APs can be very handy to salvage for ideas, even when writing your own home campaign from scratch. They sometimes have scenarios or special rules that can be adapted really well to help put together something that you want to do.
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ap.
#1 - it will save you time.
#2 - you and your players can talk to other people about their experience with the same adventure. folks not at your table will never really be able to appreciate the PERIL MEN OF MOON 5, but you can easily strike up a conversation about the Driftrock and compare notes.
#3 - Dead Suns is great, and you'll have fun. But make sure to get that 4th player, cuz it's unforgiving.
AP for the time saving. Home campaign if you have the time and energy to build one.
Though, in regards to what the others are saying about a fourth player... If all three PCs are built with a combat first mentality, I think they would be fine in most encounters for dead suns at least. That really depends on your players, but there's quite a few opportunities for helpful NPCs along the way.
APs are good for just playing without a ton of thought put into them, while at the same time being rewarding for the players. Home campaigns can be a lot more work to make but can end up being much more engaging and rewarding for both players and the GM. For a group of three, a home game is probably going to be easier to balance.