Under normal circumstances only one application is in the foreground and actually running. Of course what that app is doing can have a big effect on battery life.
Other applications may be multitasking, but only in very specific circumstances - it's rare. Most apps will actually be doing nothing, so the answer is conditionally, "no".
When you tap the home button the foreground app receives a set of messages that allow it to prepare for and then enter a background mode. The app doesn't have to terminate completely but it is not running either. By default apps may not continue running in the background in iOS. They may request permission to run for up to ten minutes in the background (for example to complete a download) when they are notified that they are about to lose foreground status. Abusing this would be one way to get an app rejected by Apple.
Fraser Speirs recently wrote a very interesting blog post about this which was widely retweeted by developers, because it is right, and current for iOS 5.x.
Quoting from it:
There are exactly five kinds of apps allowed to run indefinitely in the Background state in iOS 5:
- Apps that play audio while in the Background state. A good example is
Instacast while it's playing a podcast.
- Apps that track your location in the Background. For example, you
still want voice prompts from your TomTom navigation app, even if
another app is Active.
- Apps that listen for incoming VOIP calls. If you use Skype on iOS,
you can receive incoming Skype calls while the app is in the
Background.
- Newsstand apps that are downloading new content.
- Apps that receive continuous updates from an external accessory in
the Background.
Now if you are running an app that fits into one of the above categories it could be performing background processing. If it is a badly written app it could possibly be doing unnecessary processing. If at the same time you were running a foreground app that is demanding of processing resources then it's possible that killing the background app could make your foreground app go faster.
However! If you look at the categories you will see that these are all things that you would want to do if you were using those apps. You would not want to be able to receive a Skype call only when you brought Skype to the foreground. You would not want TomTom to stop giving you directions just because your passenger started to check email. So terminating apps is almost always a bad idea.
Now a word about the Recent Apps List that you bring up by double-clicking the home button. Notice the name is not "task list" or "multitasking list" or anything like that - it's just a list of recent apps. Some of them may be in the background, suspended and ready to run. Some may be terminated and not running at all - if tapped they would have to start running from scratch. An app from one of the listed categories may even be running in the background. But this list gives no indication of what is running right now. You can remove apps from the list but if the app wasn't running it can't be terminated. Proof: do a hard reset of your phone, letting it boot from the white apple screen. All the icons remain the same. Apps that may have been running are now not running but their icons remain.
Generally speaking there is no need to manually remove an application from the recent apps list.
If you really want to terminate an app that is currently in the foreground without switching to switch away from it before terminating it via the recent apps list, you can hold down the sleep/wake button (top of the device) until slide to power off appears. If you then hold down the home button until the device returns to the springboard, the app is terminated. (If you press Cancel and return to the app instead it is not terminated.) Proof: use an app with state - e.g. do a search in the iTunes Store app. Terminate it in this way, when you reopen the app your state is gone. If you don't terminate it your state remains.
Generally termination of the foreground app is never required - as you may have seen, an app that crashes is terminated by the phone automatically as soon as the watchdog process detects that the app is not responding to system events.
From the iPhone OS 4 press event in 2010:
Q: How do you close applications when multitasking?
A: (Scott Forstall) You don't have to. The user just uses things and doesn't ever
have to worry about it.
A: (Steve Jobs) It's like we said on the iPad, if you see a stylus, they blew it.
In multitasking, if you see a task manager... they blew it. Users shouldn't ever have
to think about it.
Best Answer
When you disregard things such as dimming the display on battery and CPU-throttling - you're left with a quite obvious difference: charging the battery.
Even at 100% charge, you'll spend some extra energy keeping the battery charged while using the computer.