@ff524 found the 2017 parade route, 2013 parade route, 2017 ticketed users, and 2013 ticketed users access points. The 2013 ticketed users screening location was in the same place as in 2017 and had magnetometers both years. @DavePhd found Congressman John Carter's inauguration page which says, "Security requires ticket holders to arrive at least a couple of hours early, carrying no seats, camera bags or other large bags. You will be required to enter through metal detectors". So Acosta's statement is simply incorrect. The portion of the Mall which was ticketed did have security screening.
The 2017 parade route shows that the National Mall had limited entry points in 2017. That does not answer the question of whether or not magnetometers were in use at those entry points in this year or 2013. Other sources suggest that they were not.
The actual parade route (not on the Mall) had essentially the same security screening in both 2013 and 2017. This does not affect the truthfulness of either statement though, as it had nothing to do with screening for the Mall.
@ff524 found an ABC News story which says
The Secret Service says that this was the first time security fencing was installed around the National Mall for an inauguration.
However, there are at least two factual inaccuracies elsewhere in their story, so take that with a grain of salt. If believed, this would confirm Spicer's assertion that there was more fencing in 2017 than 2013. However, the same story contradicts the assertion that there were more magnetometers. Unfortunately, it does so in a factually inaccurate way:
To get onto the Mall, people were required to go through one of seven checkpoints where their bags were checked, but there were no magnetometers used at those checkpoints.
But we already know that the ticketed portion of the Mall had magnetometers. The conflict may be as simple as a missing 'the non-ticketed portion of' the Mall. Or there may be a more serious problem with that story.
If we include the Washington Monument as part of the Mall, then the parade route PDF confirms the seven checkpoints to the non-ticketed portion of the Mall. Note that that does not include the eighth checkpoint for the ticketed portion of the Mall, which is shown in the ticketed users PDF.
If we believe the basic thrust of the ABC News story, Spicer was half right. There was more fencing but no more magnetometer locations.