I want to buy a smartphone below Rs 14000

Comparative analysis of all the three phones you mentioned would be helpful in reaching a good decision about which one is better.

Samsung Galaxy S3 Neo

The phone has physical dimensions of 136.6x 70.7x8.6mm at a weight of 132 grams. The device is powered by Cortex A7 1.4GHz Quad core processor.It has 1.5 GB RAM which should be enough to handle any kind of multitasking. The phone comes with an internal memory of 16 GB that is expandable upto 64 GB.
The device has a 4.8 inch Super AMOLED touchscreen at 720x1280 resolution and 306 ppi pixel density.
Galaxy S3 Neo runs on Android 4.4.2 kit kat. It may receive Lollipop update soon.Rear camera on the phone is 8MP at 3264x2448 resolution while front camera is 1.9MP 720p resolution. Battery capacity of the device is 2100mAh.

HTC 620G

The phone weighs 160 grams at 150.1x72.7x9.6mm. It is powered by Cortex A7 1.7GHz octacore processor. Device has only 1GB RAM which is a bit low compared to the phones these days. The internal memory is just 8 GB. It can be expanded upto 32 GB.
The phone features a 5 inches capacitive screen at 720 X 1280 resolution and 294 ppi pixel density.
The device runs on Android kitkat 4.4.4 with likely update to Lollipop. Phone has a rear camera at 8 MP with 3264x2448 resolution.Front camera is of 5 MP capable of HD recording. The phone battery has a capacity of 2100mAh.

Moto G2

The phone physical dimensions are 141.5x70.7x11mm with a weight of 141 grams. It has 5 inches IPS LCD screen at 720x1280 resolution and 294 ppi pixel density. The device is powered by Snapdragon 400 1.2 GHz quad core processor. The phone has internal memory of 8 GB expandable upto 32 GB. The phone has just 1 GB RAM which may be insufficient for heavy apps.
The rear camera on phone is 8 MP at 3264 X2448 resolution while 2 MP front camera is capable of 720p video recording.
The device runs on Android 4.4.4 Kitkat with possible up gradation to Lollipop.

My view

My view says it would be better to go with Galaxy S3 Neo as it has more RAM. My personal experience has made me understand that RAM is the most important aspect in addition to the internal memory while buying a phone because if you don't have enough RAM you won't be able to run the apps even if you big spec sheet. All these three devices are almost similar to each other with a little difference here and there. S3 neo has no 4G support while HTC has it. HTC scores with 5MP front camera over the two other devices. But basically any one of them can easily replace the other. However, being a pure Android - Motorola offers a perfect mix of features, performance and timely updates. It could be a best buy for a sub 15k phone. Available for Rs.12999 on FlipKart, you can opt for it.
There are other options available in this range in Micromax, Xolo and Asus, but since you want phones from the brands you have mentioned - these would be the ultimate ones with emphasis on Moto G2 and Galaxy S3 Neo.

My suggestion is Samsung S3 Neo because this is value for money smartphone. I am using this phone for a month and below are the advantages of this phone.
1. Amazing display
2. Awesome camera(Both)
3. Excellent performance
4. Flawless battery backup
5. Good quality product,you will be very happy.

One of my freind using the HTC 620G DS but the price of this phone is high. And it has below disadvantages.
1.Touch is bad: I played game in HTC 620G DS and touch is not so responsive.
2.It runs old version of android ie Kitkat 4.4.2 whereas Samsung S3 Neo's OS is Kitkat 4.4.3. Kitkat 4.4.2 has bug in music player.
3.HTC is known for design but this model has ugly look. The plastic body of the phone feels like a China mobile.

As per my opinion Moto G2 will be the best one to go for from those three devices which you mentioned above.

Motorola Moto G (Gen 2)

You can get this at Rs. 12,999 on Flipcart.com. It was released in the month of September, 2014 with the dimensions of 141.50 x 70.70 x 10.99 mm. It is 149 gram in weight. It is available in the two basic colors and they are Black and White. It comes with the medium size screen of 5.00 inches with the dimensions of 720 x 1280 pixels. It has 294 PPI which is considered to be below average. It runs on Qualcomm Snapdragon 400, 1.2GHz quad-core processor with 1 GB of RAM. It comes loaded with the internal storage space of 16 GB and the external storage space can be expanded via microSd card. It has a rear camera of 8 MP with flash and a front camera of 2 MP. It runs on Android 4.4.4 and also supports Java. It supports Wi-Fi (802.11 b/ g/ n/ ac), GPS, Blue-tooth v 4.00, DLNA, Wi-Fi Direct, 3.55 mm jack, FM, Micro-USB, Dual SIM and it can also be charged via Micro-USB. It comes loaded with three sensors and they are Compass/ Magnetometer, Proximity, Accelerometer and Ambient light sensor.

If you are interested you can buy BlackBerry Z3. The phone is currently available for Rs.13000. I am using this phone for three months and I had a wonderful experience. If you are worried about apps be advised it supports Android apps. Yes, you can install any apk files in this phone.

You will spend most of your time in browsing. It has got best mobile browser I have ever used. It has inbuilt flash support and you don't need to do any tweaks to play online videos.

Emails and messages are collected in a single screen called Hub. It is easy to manage mails and messages. If you are working in an organization then sure you like this feature.

Internet is faster than other devices. Even in 2G connection I can able to download at 17 KB per second. In usual 2G connection you will get only 8 KB per second.

Battery backup is too good. I feel like I am using a basic handset which gives generous battery backup.

I was Android user before. After changing two or three phones I feel like I was using same mobile. Then I checked for other options. iPhone price is way too high. Windows Phone was popular at that time and so I went for one. Even Windows Phone experience is good but I gave it to my mom as everyone carried one. I want to stay out of the crowd and then I chose BlackBerry Z3. I never realized the BlackBerry 10 OS is so mature.

Go for it if you want excellent battery backup, email and browsing experience.