In an IT context 実行 is execution in the sense of executing a program. 実施, on the other hand, refers generally to putting some plan into action. 実行 and 実施 do have overlapping meanings in the sense that they involve setting something in action, but the IT meaning is exclusive to 実行 in most usage that I've seen (and in the dictionary) though there may be some overlap there.
In broader terms separated from IT, 実行 is something more labored and requiring some sort of judgment or effort beyond that which you would associate with 実施, like "time to execute plan B!" I associate 実施 more with the English meaning of "conduct," as in "conduct an examination/experiment/survey/whatever." It refers simply to a real life implementation of something that is planned but that is not extraordinary in any particular sense.
There's a nice answer here that has a little more discussion about it: http://okwave.jp/qa/q7438350.html