Smaller games are simpler. The fact that there's less going on combined with the lower model count makes it easier for rookie gamers to assemble and play. These are good learning games or experiments for a new army. Espically in times like this when a new codex drops, it gives a player an opportunity to see just what new units may be capable of.
Beyond that, there's a greater emphasis on tactics in smaller games. Just the fact that you're limited in points means before the game you're forced to make very specific tactical desisions when building your army. You don't have a ton of points to bring everything so if you want to bring that big death star, be aware you may be limiting yourself elsewhere. Some bigger nasties and netlists don't even work under a certain point level, which is probably a good thing but it's there. Just remember there is no such thing as an army that "doesn't work" at a smaller points level. People that say that just don't like having to budget their points and "can't play" without Draigo and his paladins or a massive gutstar. In my opinion it's just laziness.
Small games are, by their very nature faster. This is great for people that can't always dedicate a whole afternoon or more to a game. This is also something tournament organizers take into consideration. There are 5-game 1 day events, but good luck running an Ork horde if the points are too high.
Finally small games do affect game balance in both directions. If you do bring something big and nasty, your opponent has to think of something that could counter that. Of course this could backfire if you bring a big non-scoring unit and 2 small, cheap troops that can be easily killed. If your opponent gets to them, you're basically hoping for a draw in some scenarios. On the other hand, if you do toss a ton of points into that massive unit, I just may be lucky enough to take it out and there goes most of your army. As I said, the list is a very tactical thing at smaller points.
On the other side, you have the big games. For argument's sake i'm looking espically at the games that hit that "extra stuff" threshold. For 40k thats the 2000 point double force org. and in Fantasy it's the 3000 point Grand Army. I'm not considering Apoc or Storm of Magic since these are designed to be different experiences. The best advice I heard to a new Apoc player was "don't get too attached to anything" and it's true. Apoc and Storm were designed for massive forces to get together with outrageous rules and see what happens. I'm looking at big games that run in the regular rules only for this article.
So big games...well pretty much everything I said above is the opposite. They need lots of models and lots of time. They become complex and can be a challenge to rookies. Players not used to the game can be overwhelmed by all the things to keep track of and may easily forget to do things. This also means big games get expensive, but that's why I am a huge advocate of finding cheaper 3rd party alternatives to GW's more overpriced models as long as you can tell easily what's what, but that's another issue.
You'll have a different experience building your list. I do think it's easier building a big list as it's a piece of cake to fit in those 4 Hydras or 6 Long Fang squads. Of course the balance is you then are facing a mass of Empire cannons or something equally vicious. I will say though that while all armies can work at lower point levels, there are some that don't scale up very well. Fantasy has another issue since magic no longer scales either. There are a few armies that MUST have a good magic phase and even 12 dice may not be enough. Of course sonetimes it's just fun to bring out every model you own and throw it in a massive fight.
Of course, whichever game size you prefer is up to you and both have their merits. At the end, I say play both and just have fun. After all, isn't that why we play the game?