Oh and no way I'd say the Note 3 is faster than the 5s. As much of a power house that it is, it still lags a bit here and there. It runs better on Nova, but when something stock like the dialer comes up, there comes the lag.
The Note 3 runs circles around the 5s in real world testing at our office. Especially in terms of working on or viewing 2mb spreadsheets which we use daily for our production reports. We deploy both phones and I have used both and will gladly stick with the Note 3.
In terms of the dialer....yes, the stock TW Dialer sucks. The beauty of Android however is there are dozens of different dialer apps you can replace the stock one with. I use EX Dialer. It's blazing fast, ZERO lag when you pull it up and it's way more robust in terms of features and themes. Again, even replacement dialer can be customized for look and functions.
It's not the phone that lags, it's touchwiz. Just ditch it. You'll enjoy the phone a lot more and realize that it really is way more customizable, faster and overall offers more than an iPhone.