Yes, they're different in terms of both grammar and meaning.
'Read' is what's known as an ergative verb, meaning that it has two forms: transitive and intransitive. Another example of an ergative verb is 'close'. For example, we can use 'close' as an intransitive verb in the sentence "The shop closes at 6 pm", or as a transitive verb in the sentence "We close the shop at 6pm".
When we say "Your sentence should read as follows", we're using 'read' as an intransitive verb. It means "This is what the sentence should 'say' " or "This is what your sentence should look like".
When we say "Your sentence should be read as follows", we're using 'read' as a transitive verb in the passive voice, meaning "This is how you should read your sentence". You'd say this if you are giving someone tips on pronunciation, or on delivering a speech or presentation.