The LG G3 has HD Voice where the network supports it and you can enable a number of features to make the audio even better. Noise suppression will reduce background noise from your end, while Voice enhancement lets you hear the other side more clearly.
For HD Voice you can also select the Clean or Soft presets to modify the sound. There's also an option to display a Record call button during calls.
Additional options let the phone block certain numbers from calling you or have this apply to all private numbers, it can also block international calls and certain outgoing calls (e.g. block 700 numbers). Privacy keeper will hide the person's name or all identifying info if you're worried about someone peeking at your screen when you get a call.
Quiet mode can turn off sound and light notifications on a schedule so calls and messages don't disturb you. The corresponding HTC feature is called Do not disturb and can be scheduled too. There's a list of exceptions to make sure important contacts will always get through.
The LG G3 messaging app can block unknown numbers too or messages coming from specific numbers or containing specific phrases. Text Link is a really cool feature that links certain apps to related message content. For example, getting a text about a meeting will bring up the calendar icon so you can easily add an event. A message with an address will link to Maps instead.
The HTC One (M8) also has blocking functionality for calls and messages based on the caller/sender number. It can also disable dialing home numbers when you're roaming.
For messages the One (M8) also has a Secure box - it locks away selected texts with a password. Those can later be deleted or moved back to the general inbox.
The HTC flagship utilizes Polaris Office 5, a full-blown Office package. It can view/edit/create Word, Excel and PowerPoint docs and view PDFs. An interesting option lets you export a document as a PDF (all desktop browsers can open PDFs but not .DOCs). You can also annotate the documents with hand-written notes, which also go into the PDF.
The LG G3 meanwhile scrapes by with an app that can only view documents but not edit them.
The South Korean flagship comes with both Chrome and an LG-modified browser. That one has a couple of features missing in Chrome - you can save pages for offline reading and you can also capture the whole page as an image you can crop or edit with an app (say the gallery).
The HTC One (M8) also comes with Chrome and a modified stock browser, but HTC only tweaked the appearance of the browser to match the rest of Sense UI.
LG Health is an answer to Samsung's S Health. It's not quite as advanced but it can still track your walking, running, cycling, hiking and inline skating. It lets you create a profile with your basic characteristics and set yourself goals. You can view charts with calories burned and distance traveled and compare your results with others in the motivational Weekly ranking screen.
HTC has a similar app borrowed from Fitbit (it's free, you can download it from the Play Store). Besides distance traveled and calories burned, the app can also keep track of what you've eaten and how much water you drank. Fitbit also has a sleep tracking feature, but note that some of the app's features rely on you having a Fitbit device.
HTC Backup can save apps, contacts (including linked social accounts), mails and text messages. The data is sent to Google Drive for safe keeping.
The LG G3 has a Backup function, but that's hidden in the Settings menu. It stores app data and settings to Google again and will automatically restore the settings of an app you reinstall. You can also schedule local backups with this feature.
Both phones offer to pull data from your old phone, easing your transition to your cool new flagship.
Another cool tool is HTC Guide. It provides useful info that customer support might ask you about right on the Contact Us screen. It also lets you do your own diagnostics (e.g. test the buttons or the light sensor), review power and mobile data usage and monitor the phone's storage.
The Tasks app on the HTC One (M8) lets you create multiple separate lists, including ones hosted on cloud services like Google Drive. It also lets you add a location to each task so you can view on a map whatever there is to do in your area.
LG's Tasks app also supports multiple lists.
Winner: HTC One (M8). LG offers a rich app package out of the box, virtually equal to the HTC in all but one respect - there's no Office document editor. The Polaris Office 5 app to the HTC One (M8) is very capable and enough to tip the scales. HTC Guide is a great idea for non-technical people too.