A restricted passport is most commonly issued as a second passport when the traveller's main passport has ceased to be valid for travel to specified countries.
For example, some Arab states won't admit travellers with an Israeli stamp in their passport. If a UK passport holder, who has already visited Israel, wishes to visit such an Arab country, he can apply to be issued with an additional passport. However, this passport would only be valid for a short duration and would only be valid for entry into one or more specified countries. (A restricted passport can only be issued for entry to a country with which the UK has diplomatic relations).
I've written all of that in the present tense but there's nothing on the Passport Agency website about restricted passports, so I'm not sure whether they're still issued.
Chris