The difference between tank, APC, IFV, Tanks Destroyer/SPG is intended/designed role.
A tank (MBT these days) was intended to fill the role of infantry support, break out and exploitation (smashing holes in the front line and causing a mess in the enemys rear). Since the end of WW2 days they also fulfil the role of the old Tank Destroyers due to improved firepower (e.g. the M26 Pershing had the same 90mm gun as the M36 Jackson/Slugger TD).
A Tank Destroyer was intended specifically to engage and destroy tanks and other armoured vehicles. For instance the design philosophy behind the M10 and M18 TDs was that they were lightly armoured to improve speed so they could rapidly deploy from rear areas to stop enemy armour breakout. In reality of course they were often used in the same roles as a tank. Other nations TDs were basically existing tank chassis with a new hull to mount a bigger gun than could be mounted in a turret on the same chassis. A good example of that would be the Hetzer: an old, obsolete Pz38t light tank turned into an effective light tank destroyer.
Self-Propelled Guns are exactly what they say on the tin (Tank Destroyers are just another type really). Either mobile artillery (M7 Priest, Sexton, Hummel) or AA (Wirblewind, Crusader AA, M42 Duster).
IFV are an evolution of the APC (which is just an armoured box for keeping troops with the tanks really), capable of providing fire support to the embarked infantry squad as they dismount. Their armament tends to be lighter than MBTs as they are not intended to engage enemy heavy armour (plus there would be less room for infantry). However against older MBTs they can still be effective (as seen in the Gulf War, M2 Bradleys taking out T-55s, Type-69s and T72s). You wouldn't want to pit an M2 Bradley or FV510 Warrior against an M1 Abrams or Challenger 2 though
Most people however view anything with armour, tracks and a big gun as a "Tank".
just look at the Cromwell-tanks: they would practically stop functioning when a panther or Tiger looked at them in a funny way, and the German tank crews just laughed when they opened fire.
I know it's a late reply to this but seriously...
Armour:
Cromwell - 3'' (76mm) front (later increased to 4' (102mm)
Sherman - 2'' (51mm) front (often increased in the field by salvaging plates from knocked out Shermans)
Weapon:
Same calibre and ammunition but barrel length was less. The only direct comparison (same range and angle of attack) I have between the ROQF75mm and M3 75mm gives a 2mm penetration advantage to the M3 75mm. 76mm armed Shermans were not used initially in Normandy (started to be deployed in July 1944. Pattons 3rd Army didn't accept it until September) and still only made up half of the Shermans in Europe by the end of the war.
Speed (on road):
Cromwell 40mph (unrestricted. later restricted because the tracks would tear off when cornering too fast).
Sherman 30mph
The Cromwell also had a lower profile making it harder to hit. Not very numerous though, only one division (7th Armoured) was fully equipped with Cromwells.
So not a complete joke like you seem to think it was! It compared pretty well to the Sherman and was just as much a threat (or not as the case may be) to the German tanks as the M4(75mm version). Granted the Sherman had one major advantage - unlike the Cromwell it could be up-gunned as well as easily modified to fulfil specialist roles (e.g. Hobarts Funnies). The Cromwell, like all British tanks of WW2 (that saw combat. The Centurion missed it) suffered the oh-so-stupid design limitation to make it fit on existing transport trailers. Narrow hull = small turret ring = small turret = no room for bigger guns. If only they had thrown that limitation out earlier... the Cromwell could have been Centurion mk1.
(note: Military history, particularly WW2 Armour, is an interest of mine. I won't say I'm an expert and I will stand corrected if someone has greater knowledge/expertise than me. I just don’t like some peoples view of WW2 Britain = Suck, US = Greatest Army EVAR!!!!)
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