It's entirely dependent on what you mean by 'Anglo-Saxon warrior,' honestly. If you're referring to one of the semi-professional warriors who would accompany local nobility, then there'd be a huge amount of difference: They'd be, generally, far better armoured than their viking counterparts (who, you have to remember, were essentially the biker gangs of the North Atlantic) and typically carrying better quality weapons than your average Björn-about-the-fjörd viking.

That said, the weapons themselves wouldn't have much of a variation; spears were the most common weapon of the day - and for the Anglo-Saxons, they held a very high cultural place, given the appearance of the '-gar' element in so many names, not to mention its appearance in loads of poetry - followed by axes, daggers, and finally swords. As far as defensive accoutrements are concerned, you'd be looking at (typically) round shields for the Anglo-Saxons, and a mixture of round and kite shields for the vikings (kite shields really became a thing in the 9th century, and they showed up all over western Europe, which happened to be right where vikings would raid, so they'd likely have picked some up) Helmets were typically made of iron, though leather could have been used. Mail was expensive and wouldn't be particularly common among raiders, but wealthier Anglo-Saxon warriors likely would have owned a byrnie, if not a full length hauberk. Other armour could have included boiled leather or thick cloth; scale armour, as was suggested by /u/Whatsthedealwithair-, is practically unheard of outside of Sweden (and then it's one single example) and not mentioned in sources until the 13th century (a reference to a spangenbyrnja, or plate shirt, which could be a reference to scale, lamellar, brigandine, or plate armour, we're not sure) so that's not likely to have shown up.

Otherwise, apart from quality of steel and iron, they'd be using basically the same styles of weapons; the constant interaction and cultural connections between the Anglo-Saxons, Danes, Norwegians, Swedes, and Franks led to a sort of homogeneity of arms and armour design, which is reasonable considering their proximity and constant interaction.

Almost none. Both would be wearing little armour beyond a leather or metal helmet, unless they could afford chainmail or rarer still, scale. Both would typically carry a round wooden shield with a metal boss, later kite shields were used by both cultures. Spears were the most common weapon, axes were widely used too whilst swords were, like mail, confined to the well-off. Javelins were the predominant ranged weapon for both cultures.

Neither culture placed much importance on either cavalry or archers, the Anglo-Saxons in particular having very little record of archery in a military context. Both of their forces would be almost entirely melee infantry, with cavalry used only to scout and pursue fleeing enemies.

They are so similar because of their origins, the Saxons originally came from the area that is now Northern Germany/Southern Denmark. Their style of warfare was almost identical to the neighboring cultures to the North.