We all saw that after Vito Corleone stepped down as Godfather, Micheal turned Tom out as consigliere, but I see Tom is still active as Corleone family adviser, so what difference came after Vito in Tom's role?
In layman's terms, Vito didn't see Tom Hagen as being ruthless enough. Tom was more "on the level", very diplomatic and knowledgeable of law (being that he was a lawyer). In "wartime", when families are fighting, diplomacy isn't the norm. You need a Consigliere who works under the premise that sometimes you need to be ruthless and call for hits (murders) of other family members and/or heads.
Vito (and Michael) had an enormous amount of respect for Tom. They trusted him implicitly, which is why they retained him for other advice. Once the war was over, Tom was reinstated as Consigliere.
The real difference for Tom's temporary "demotion" was that the Corleone family was at war...and Tom was not a wartime consigliere in the opinion of both Vito and Michael.
From the Godfather Wikia ( thanks to BCdotWeb for the link).
After Michael Corleone became operating head of the Corleone family, he removed Hagen as consigliere on his father's advice, restricting him to handling the family's legal business in Nevada, Chicago, and Los Angeles. When Tom asks why he was being removed, Michael answers by telling him he just isn't "a wartime consigliere", adding that it was no reflection on the work he had done for them in the past.
Despite his reduced role, Tom worked closely with Michael in planning the massive slaughter of the other New York dons. Immediately after Michael formally becomes the Don (following his purge of the other Families) Tom is reinstated to his former position.
More reading of the deep trust that Michael continued to have in Tom (including making him acting Don at one point) at the Wikia link.