Why the contrasting results? The political composition of samples doesn't appear to offer many answers, and that wouldn't really make sense anyway, given that Obama's job approval rating was slightly better in the WaPo-ABC poll.
Instead, as with many things in polling, it depends on how you ask the question. While the WaPo-ABC question on blocking Obama’s action asked about the policy in a single question, CBS's question on blocking policy was preceded by a question about support for the policy overall, which was worded as
The polls differ both in how they describe the number of undocumented immigrants affected and the specifics of deportation relief. The CBS survey said the order would allow "some" illegal immigrants in the United States to stay, while the WaPo-ABC poll referred to "as many as four million."
In addition, the WaPo-ABC poll described the policy as allowing undocumented immigrants to "avoid deportation," while the CBS survey said it would allow them to "stay here temporarily and apply for a work permit if certain requirements are met." Before asking whether Congress should block the policy or let it go ahead, CBS found 62 percent favoring the proposal as described -- a basically unprecedented level of support.
If Americans are indeed more wary of legal status to a very large number of immigrants, the more specific "four million" wording may have limited support in the Post-ABC poll, rather than the “some illegal immigrants” in the CBS question. The specifics offered in the CBS question might have also boosted support for the policy, and reduced support for Congress blocking the proposal. Polling has regularly shown that the more hurdles mentioned in the poll, the higher level of support for a path to legal status.
Specific requirements have been connected with varying support for immigration in the past. Between mid-2013 and early 2014, support for a path to legal status or citizenship ranged from 46 to 81 percent in public polls. Surveys that included more requirements -- such as paying back taxes or passing a background check -- tended to show higher support for legal status, with a positive statistical correlation of 0.74 on a scale of 0 to 1.
The divergent results confirm that specifics matter quite a lot in public opinion on immigration, with opinion nuanced in ways that can benefit or hurt both Obama and Republicans. Americans' general reaction to allowing undocumented immigrants to stay in the country is negative, but support rises to a majority when the situation is temporary, dependent on working or involves other requirements. Tactics of both sides appear palatable as well; Obama's use of executive action has not been broadly rejected, but neither are Republicans' ideas about defunding department budgets to block it.
Both parties also have to worry about pleasing not just the electorate overall but Hispanic voters in particular, for whom immigration is a top-tier issue. Hispanics have expressed support for Obama’s executive action regardless of question wording in WaPo-ABC polls from December and January. And surveys have tracked a sharp rise in his overall job rating among Hispanics since he announced the new policy on deportations, indicating Republican challenges could be received poorly among this group.
Going forward, the divergent poll results are instructive. If Americans come to see Obama's actions as providing a fair deal of deportation relief for undocumented immigrants who jumped a series of hurdles to stay, it could prove popular over time. But if those requirements are seen as insignificant or easily skirted -- as Americans have viewed the nation's border protections in the past -- the opposition could win the day.