annihilate多用于战争中的歼灭，毁灭性的打击等等，而eradicate用于消灭一种现象、行为或多余的东西。
1、annihilate
an.ni.hi.late
v.（动词）
an.ni.hi.lat.ed， an.ni.hi.lat.ing， an.ni.hi.lates
v.tr.（及物动词）
(1)
To destroy completely:
歼灭，毁灭：彻底消灭：
The naval force was annihilated during the attack.
在这次袭击中海军被彻底消灭了
(2)
To reduce to nonexistence.
使消失
(3)
To defeat decisively; vanquish.
彻底击败；战胜
(4)
To nullify or render void; abolish.
使无效，使废弃；取消
v.intr.Physics （不及物动词）【物理学】
To participate in annihilation, as do an electron and a positron.
湮灭，湮没：湮灭，如电子与正电子
2、eradicate
e.rad.i.cate
v.tr.（及物动词）
e.rad.i.cat.ed， e.rad.i.cat.ing， e.rad.i.cates
(1)
To tear up by the roots.
根除：连根撕碎
(2)
To get rid of as if by tearing up by the roots:
清除，扑灭，根绝：象连根拔起一样消灭：
Their goal was to eradicate poverty.See Synonyms at abolish See Synonyms at eliminate
他们的目标是消除贫困
