Actually, I would say that Google's apps run better on iOS than Android. I've been using Android since the OG Motorola Droid in ~2009, and I just switched to my first iPhone (6s Plus). Of course, being a long time Android user, I use Google's services extensively. One of the first things I noticed is how much smoother and more responsive Google's apps are on iOS.
Take Google music for example, it would randomly skip songs, fail to stream, stutter, not register button presses, etc on my Moto X. On my iPhone, it's flawless. Inbox (for Gmail) is also much faster on iOS. The biggest thing I noticed is Google Authenticator, which has more of a material design theme on iPhone than it does on Android! I don't know what kind of "integration and background processing" I'm unaware of, but I haven't noticed any detriments using Google apps on an iPhone vs Android phone.
My Moto X and iPhone 6s Plus do pretty much the same things, but in general the iPhone does them faster/better. The camera is leagues ahead of my Moto X (which, to be fair, is not anywhere near the best camera Android can offer), and the battery life is incredible. I unplugged my phone yesterday at 7AM, and today it's still at 40%. That's ~29 hours with regular usage (battery menu says 6.5 hrs usage, 29 hrs standby). No Android phone I've ever used can touch that kind of battery life, unless I put it on airplane mode.
The one thing I really dislike about iOS is the sharing menu. For example, on Android, if I'm in my photo gallery, I just click the share button, and I can send the photo/multiple photos to any app I want (upload to dropbox, send via hangouts, upload to Instagram, etc). On iOS, sometimes I can share direct from the gallery (messages, Twitter, etc) but it depends on where I want to send it. If I want to post to Instagram for example, I have to leave the gallery app, open instagram, click on the camera, and then once again find the photo I want to upload. There's no option to add Instagram to the sharing menu!