"You are blocked from following @Name and viewing @Name's Tweets. Learn more"
It looks like you can't view their tweets and suggests that if you block someone they won't be able to see your tweets either. This is incorrect.
If someone's tweets are public then anyone can view those tweets by using any of the following methods
(a) using a different browser
(b) logging out of the blocked account and using the same browser while logged out
(c) logging into a different account
(d) using a different smartphone app - at the moment Echofon for iPhone lets me see the profiles of a couple of homeopathy quacks who've blocked me
(e) search for their tweets (or if they're using a hashtag you'll still see their tweets there) on desktop Twitter or with Hootsuite / Tweetdeck.
If you want to ensure that someone cannot see your tweets you'd need to
(a) make your account private and
(b) be careful who you allow to follow you (is it them under a different name?)
(c) hope none of your followers get phished or hacked, or leak your information
Remember that anyone can see tweets sent to you (by searching for to:yourname or @yourname on Twitter dot com, search).
What does a block do then?
It's acts as a mute, it's really nothing more than that.
If you block me any tweet I send you won't show up in your timeline or mentions, so I am effectively silenced. You'll be able to go looking for my tweets if you want to but they won't be delivered to you.
If I look at your profile I'm told I can't see your tweets but I can use the methods above to see them, as you can if I block you.
Blocking someone doesn't ever stop them from seeing your tweets. It just means you don't have to see theirs if you don't want to (and they can't favourite or retweet you - well they can manually retweet you).
You can see all the accounts that you have blocked at https://twitter.com/settings/blocked
Read more about private Twitter accounts
Read more about Twitter blocks
What happens if you block someone on Twitter? What happens if they block you? (7 June 2012, updated 4 January 2014 and again today (12 December 2014).