Have to go with ones I've used so there will be some (namely the Note 3) that are left off purely because I can't make an informed judgement.
Honorable Mention: Moto G/Lumia 520
Ok so I haven't owned these (yet), but given the internals being packed in and the price points, I really had to mention them. I'm hoping to pick each up here pretty soon to appease my Android and WP8 curiosity.
(5) Galaxy S4
What it does right: The Galaxy S4 is a nice smartphone - fast, well designed, with some "unique" features. What REALLY set it apart in my mind was the size-to-display ratio. As a 5" phone, it didn't feel overly large. The bezels were perfect and really what I hope most smartphones move to (not that "no-bezel" crap that's just impractical).
What it does wrong: TouchWiz. Enough said. I also am not a fan of protruding cameras which give an uneven lay on a flat surface.
Summary: Great phone with a surprisingly nice design and the perfect size-to-display ratio that unfortunately lets crappy software ruin what could be an otherwise great experience.
(4) iPhone 5C
What it does right: THIS is what a plastic phone should feel and look like. Honestly, I'm so enamored with the design of the 5C, I honestly feel Apple should design future iPhones more like the 5C than the 5S. I'm much more comfortable with a 4" device and this is the most comfortable phone I've ever held/used.
What it does wrong: I know we were all hoping the price point would come in a little lower (or a lot depending on your level of delusion). I also know I'm going to get some flak for including a device running 2012's internals on this list.
Summary: What the iPhone 5 SHOULD'VE been. If priced $200 lower off-contract, this would've been arguably the greatest smartphone release of the year.
(3) HTC One
What it does right: The HTC One was my first foray into skinned Android and I have to say Sense is by far the best of the ones I've used or seen. Simple, elegant and not intrusive. The design of the phone itself is absolutely superb and the best of any Android phone out there. And holy cow....those boomsound speakers. I wake up at night missing that amazing sound!
What it does wrong: Agh that camera....HTC was really that close to making the perfect Android handset (non-stock).
Summary: Arguably the best Android handset of 2013 didn't get the attention or pub it deserved and HTC is on the brink of failure. Beautifully designed, AMAZING sound from those speakers, an elegant skin - but unfortunately a botched camera. The HTC One is very closely my choice for the best Android smartphone of the year.
(2) Nexus 5
What it does right: There really isn't anything bad I can say about the Nexus 5. Price Point, power, battery, LTE, camera, display quality - all on par or top notch. The Nexus 5 takes the crown as top Android smartphone of 2013 in my book because of its quick updates and newer software. Stock is the way I will likely go with ANY Android device I purchase in the future - be it Nexus or the GPE devices which I haven't had the pleasure of trying out.
What it does wrong: The ONLY thing I didn't like about the N5 was the protruding camera.
Summary: With quick (for Android) updates, top of the line hardware and a price point well below other top handsets, the Nexus 5 takes the crown as best Android handset of 2013 - but for my needs falls just short of my overall 2013 winner.....
(1) iPhone 5S
What it does right: What is there to say about iOS. Stable, elegant, consistent. The ecosystem comes into play here as much as the device itself. But the device is no slouch either. The fastest handset of 2013 and with only a dual core processor, the 5S reminded us Apple isn't just resting on its laurels. TouchID is brilliant and adds security and convenience in one feature. And while we haven't yet experienced the true potential of 64-bit arch, pundits agree this release turned the smartphone industry on its head again. Beautifully designed with, I think, the most visually appealing design, the iPhone 5S is the best S release yet.
What it does wrong: While the design is visually striking, it leaves a bit to be desired where comfort is concerned. Other gripes have less to do with the device itself and more to do with the new software it runs - iOS 7 - not QUITE being ready for prime time.
Summary: Some may say the device's display size is a negative factor, however I disagree. In another thread, we may discuss the merits for larger iPhone options (for which I am a big supporter), but that doesn't negate the sheer awesomeness that is the iPhone 5S. As more and more people were saying Apple didn't care about specs, the 5S was released and blew everything away. With new innovative features in tow and the most visually stunning design in the industry, the iPhone 5S leads the way for Apple as its simplicity, consistency and power continue to outpace every other smartphone available.