I've seen and been in many games where the team would ace an enemy team and go straight for the inhibitor. This sometimes resulted in one of them getting picked off after or the team getting aced right back. Other times I've seen many teams rush a Baron right after the ace, when they could have easily gone for the Inhibitor.
Taking down the inhibitor provides map control and the ability to do baron later. Also, sometimes it's possible to kill the inhib and THEN go baron, which is always a better choice over baron and then inhib right after.
Many times after an ace, your team is low or missing people and doing baron is risky. Just make sure that if you don't baron after an ace and you choose to kill a tower / inhib, then go heal in time to a.) not get caught b.) buy and heal c.) make it back to stop enemy from baroning.
This is the question that plagues every single profressional team in the game. Several games have been lost simply because of a wrong call on doing this. IF your team is alive enough to make sure you can get it down before 3 of them spawn, then go for it. If its not, take baron and heal.
The fatal flaw for taking an inhibitor is that if you stay to long, you either get caught and die, or they just take a baron in revenge and push your towers with the buff. MAke sure you take the inhibitor and heal fast, and rush to baron. That way they can't just take baron in revenge for your push.
If you can't garuntee you cant take it fast enough (1-2 people alive) then baron might be better. In some cases, its even better to just back and heal. Take the gold advantage that you got, and do it again.
You have to judge each situation on a case-by-case basis. The ideal situation is to ace them at a base tower, kill the tower, then the inhib, then go down baron before they respawn.
However, if you ace them at a lane tower in top or mid and their respawn timers are all ~30 seconds, it will be faster and stronger to go snap up baron, then use the gold influx from aceing their team + baron to get an even bigger advantage for the next team fight.
Sometimes, however, going for baron is a mistake. If you can push to win the game before they respawn, that should always be the choice. If your ad carry is farmed and alive after a teamfight, and their nexus towers are open, it might be smarter to try and just finish the game.
On top of all of that, you have to pay attention to team hp and who is alive on the enemy team. If you kill 3 of their players and get out with something like 30% hp across your team, doing baron is usually a poor decision because their two guys can heal then contest baron by themselves. This can result in a poor teamfight for your team, wherein your allies are all low hp and the enemies come in with fresh hp, mana, and new items. Winning a teamfight in their base, only to get ace'd at baron will cause you to lose the upper hand and the game in a hurry.
tl;dr: Be conscious of which enemies are alive, the dead enemy's respawn timers, your team hp, and where you are on the map. Analyzing these variables will usually allow you to reach a quick and logical conclusion on which target to prioritize.
It depends really on the situation. If you can get the inhib easily, getting an inhib can cause them to have to worry about super minions while you get baron afterwards. However, if you're worried that the enemy may rush baron right when they respawn, then getting baron then going b might be the better choice, if no one is low enough to die from Baron itself.
Depends on what stage the game is at, how many defenses does the enemy team have left, how many allies survived the team fight where you aced the enemy team.
If you only have one or two players that survived the teamfight, it is better to push and take inhibitor, heal and prepare for a teamfight at baron when everyone respawned and is full health. This way you know your lanes are pushing in and that can give you an advantage with baron, especially if you can stall out a teamfight at baron.
If you have three or more players and very few of the enemy's turrets are destroyed, then definitely go for baron, assuming your main dps is still alive. The reason is simple, with a baron buff advantage, you can easily take any tier 1 tower with no effort. You can even take tier 2 towers pretty easily if your team can coordinate properly. In other words, a single baron buff is worth all tier 1 and tier 2 towers.
If the enemy team's inhibitor is exposed (inhibitor turret is already destroyed), then I would say destroy the inhibitor and destroy the nexus before the enemy respawns if there is time. Only really takes 30 seconds to destroy inhibitor, two nexus towers and nexus if your whole team is alive.
I asked a 2k player this same question. He told me to get the inhib always. If it is guaranteed. this is because with an inhibitor down if they don't defend their nexus minions will push this allows you to get baron anyway, So a better question would be when is it better to push in general over baron. And that is only if you can make it to the inhibitor. Baron basically gives you all outer turrets free.