Mute and block allow you to customise what you see from other accounts. They have no impact on what others can see of your account though.
Blocking someone on Twitter can never stop them from seeing your tweets (they can log out, use search engines, another account or a third party app to view your tweets).
The block function is pretty absolute and stops all communications from a blocked account. The effect of the mute function varies depending on whether or not you’re following the person you’re muting.
Muting someone that you follow
This will stop their general public timeline tweets from appearing in your home timeline. Any tweet they send that doesn’t include your @-name will disappear, but they can still send you mentions and if they RT or fav one of your tweets you’ll get a notification*. They won’t know you’ve muted them, though I suppose they may be able to work it out if you never comment on any of their other tweets. You can see who you’ve muted here (on desktop Twitter, link won’t work if you’re not logged in)
Muting someone that you don’t follow
It’s surprisingly powerful. They’ll completely disappear – they can’t send you a tweet (well they can but the tweet will never arrive and you won’t see it unless you look at their profile or search for tweets sent to your name) and while they can RT and favourite your tweets you’ll never get a notification for those either.
Muting an account that you don’t follow you is a very nice way of blocking them without blocking them. They won’t know that they’ve been muted (whereas they can find out if they’ve been blocked). This may be useful if you don’t want to get into a tedious ‘why have you blocked me’ discussion.
You can still keep an eye on their tweets
Of course you may actually want to see tweets that are sent to you, or activity on your tweets, eg to to check what someone’s up to. You can search for your own username on Twitter and tweets sent from muted accounts will show up, you can also view their profile and see what they’ve RTed / favourited etc.
Blocking has the same effect if you follow them or not
Once you’ve blocked someone their tweets won’t get through. If you were also following them then you’re instantly no longer following them (and if they were following you they’re now not following). However the big difference is that, if they’re using official Twitter apps or platforms, they can work out that you’ve blocked them. They won’t be able to RT or favourite your tweets and they may be told that you’ve blocked them, and they can’t see your tweets. If they’re using third party apps such as Echofon or Tweetdeck etc they won’t necessarily notice. Although your tweets aren’t likely to be sent to them (it may depend on the app) even if they look at your profile they’ll see all your tweets and so if they’re using those types of apps exclusively they may not realise they’re blocked.
If you’ve blocked an account but want to view some of the tweets you may need to click a ‘view tweets’ option to view without unblocking (this is app-dependent). You can log out too of course and view any public account’s tweets. Note that if the person you’ve muted (or blocked) has a very new or high-spam account then its tweets may not appear in search, see ‘Why am I missing from search?‘ (from Twitter’s support pages).
Muting and blocking achieve more or less the same aims but muting does it in a much more nuanced way. If your account is public then you can’t stop anyone from seeing your tweets but you can control (to some extent^) what you see from them, and muting achieves this without drawing attention to itself.
A great thing on desktop Twitter (I’ve not checked to see where else it is) is the option to see only Notifications+ Mentions, or Mentions from people that you follow. This dramatically reduces annoyance from people or ‘bots’ that are aggressively favouriting your tweets and means you don’t get a notification when they RT something. I’m assuming you’ve already switched off any notifications arriving by email but if not you can find the settings option (on desktop Twitter) here:
Here are the links for Notifications + Mentions, and Mentions only filtered by people that you follow – these links will only work if you’re logged in to desktop Twitter of course.
Notifications + Mentions: https://twitter.com/i/notifications?filter=following
Mentions only: https://twitter.com/mentions?filter=following
If you are getting lots of annoying tweets from people, or bots favouriting stuff then ‘muting at notifications’ level is probably a better bet as it will wipe out all tweets and activity except from people that you follow, and saves you having to take an action on multiple individual accounts.
^Seeing tweets of someone you’ve blocked
This can happen when other people that you follow retweet their tweet. What’s happening here is that (a) the blockee’s account is public (b) you follow someone and see all the tweets that they send (c) they ‘forward’ (by RTing) a blockee’s public tweet and so you’ll see it.
I don’t know of a way around it beyond switching off RTs from the person you follow (but someone else you follow could also RT a tweet from someone you’ve blocked, so it’s a bit leaky). If anyone knows a solution…