Not at all. Strike fighter, multi-role fighter, and fighter-bomber are interchangeable names for the same thing. It is just different countries calling them different things at different times. Or some marketing company trying to make them sound special in the advertisements.
There actually has been some changes, with the fighter-bomber descriptor pretty much fallen into disuse. In terms of strike fighters and multi-role fighters, there is a bit of a difference.
A strike fighter most often is kitted out for a strike mission, but also with some air to air armament, enabling it to do at least limited self-escort for a strike mission.
With multi-role fighters, the emphasis is more on being able to rapidly re-role the aircraft between sorties. Initial sortie could be to achieve air supremacy, then land, refuel and re-arm. This time, the kit is for a SEAD/DEAD sortie. Again, land, refuel and re-equip. This time though, targeting pods are included to allow red force detection and relay of information to blue forces.
If one looks at the aircraft in the US inventory even from the 1980's or 1990's, there were many instances of fighter aircraft, then dedicated attack/strike aircraft as well. As time progressed, and upgrades were completed on the newer series of fighter aircraft, they (F-15, F-16 and F-18) were kitted out with systems which used to be found on the A-series of aircraft. Part of the reason behind this change, was that many of the A-series aircraft (A-4, A-6, A-7, etc) were subsonic attack jets designed in the immediate post-Korean War era and many had seen service in Vietnam. As such, the aircraft were not new, which means as airframes age, they become more difficult and cost intensive to maintain, they can also become more difficult to keep upgrading. By adding more capabilities to the newer designs like the Eagle, Falcon and Hornet, then older aircraft can be phased out and replaced with new models of aircraft already in service. This rationalization of fleet inventory can make training personnel easier, and reduce the burden on maintenance. Another advantage from doing so was that the newer fighters were much more capable aircraft than their predecessors, between the addition of equipment for attack and strike missions, plus the already designed in air combat radars, modern multi-role fighters began to appear. This eased the burden on the system since the current aircraft had a reduced need for fighter escorts during mission sorties, again making things more efficient.
The basic roles still exist, but what has changed in a number of instances, is that smaller numbers of aircraft and designs are required to fufill these roles.
-Cheers
Actually with history repeating itself. Here we are in 2011. The free world is in ecomnomic collaspe, Multiple free democratic nation's are cutting their defense budgets, defense R&D, and putting all the eggs in one basket. EXP; one type of ship, one type of aircaft, etc, etc.