If you want the best specs VS battery life, get the Note 3. I owned an S4 and while it was a great device (I don't know why you heard that the battery was bad...it's not, but the Note 3's is better) I missed the S-Pen dearly, coming from a Note 2.
The S5 and the Note 3 are pretty much the same with different screens. Of course the S5 has a faster processor since it came out later, but in 'real life' performance, you'll hardly notice any difference. The screen is another factor to consider, and of course, the overall size of the device. Both are pretty big if you compare it with, say, an iPhone or a Galaxy S3.
With the S5 you get water/dust protection (IP67) rating, so you can submerge it in up to 1m of water for 30 minutes (underwater pictures/video! yei!). With the Note 3 you get the S-Pen. The S4 is really 'last year's' version of the S5, so you get lower-specs processor and a slightly smaller screen (5 inches VS 5.1 inches on the S5).
All three phones have a full HD resolution, so the bigger the screen, the less PPI it has. But all three screens are gorgeous to look at.
I've been using the Note 3 for almost a year now and I can tell you that I'm very happy with it.
Also, if you're going to buy one of these phones, you MIGHT want to hold out for a couple of months. Why? The Note 4 is just around the corner, as is the iPhone 6, and that only means lower prices for older models (or if the Note 4 calls out to you, yet another option to consider). Just a thought to save you a few bucks, but the trade-off is that you have to wait a couple of months.