That's not really true, although Trump keeps using that as an excuse. Some articles about the differences are linked below, but here are some quotes from the articles:
From politifact: "Separations under past administrations rare, systematic under Trump. Generally, a child and an adult who arrive together at the border can be separated when border officials cannot establish the custodial relationship; when they believe the custodian may be a threat to the child; or when the custodian is being detained for prosecution.
Immigration experts have told us that family separations were relatively rare under Obama and other past administrations. They did not happen at nearly the scale that they did under the Trump administration."
From vox: "It is true that migrant and asylum-seeking families were separated while Obama was president, but only in extraordinary circumstances...During the Obama administration, family separations were rare and predicated upon two conditions: whether border officials felt the parents or guardians posed a threat to the children, or whether the adults, under U.S. immigration law, had to be detained based on prior criminal convictions. While Obama was undoubtedly tough on immigration — his administration still holds the record for most deportations — border officials used discretion during his presidency to determine which illegal crossing cases to prosecute.
On the other hand, in April 2018, the Justice Department under then-Attorney General Jeff Sessions instituted a “zero tolerance” policy that called for every illegal entry case to be prosecuted. This resulted in children being separated from parents — even when the parents had done nothing more than try to cross the border.
It’s not that no family was ever separated at the border under the Obama administration. But former Obama administration officials specify that families were separated only in particular circumstances — for instance, if a father was carrying drugs — that went above and beyond a typical case of illegal entry. We don’t know how often that happened, but we know it was not a widespread or standard practice.
Under the Trump administration, though, it became increasingly common. A test of “zero tolerance” along one sector of the border in summer 2017 led to an unknown number of family separations. Seven hundred families were separated between October 2017 and April 2018." So, yeah, there are some definite differences.
Link #1:
https://www.politifact.com/truth-o-meter/statements/2019/jun/21/donald-trump/donald-trump-again-falsely-says-obama-had-family-s/
Link #2:
https://www.vox.com/2019/6/21/18700575/trump-family-separation-policy-obama-time-telemundo
