Both. That an atheist may be less likely to interpret the experience through religious-flavoured symbology, and so have a clearer picture to interpret and to integrate into their lives
There are some problems with this idea. Let's put it into these terms instead of atheist versus religious. Think of this in terms of rationalist frameworks versus mythic frameworks, particularly so speaking in terms of modern times as it seems clear we are based on the context of the discussion. Bear with a minute and see if this proves helpful.
Regarding rational frameworks, I mean thinking in terms of modern explanations of the world. How we translate the world is through looking to data and empirical evidences, logical deduction and whatnot. It's not that people who operate using mythic structures to translate the world are not rational, it's just that the interpretive filters they use to translate the world through, including their experiences, are based on mythic structures, such as deity symbols, the gods as higher forces working behinds the scenes, etc. The rational structures are science and reason and whatnot.
To put a point on this, you can have religious people who are clearly using rational structures as well as atheists. So it really isn't a case of atheists versus religious in how they interpret and integrate mystical experiences (assuming spontaneous peak experiences for this discussion, versus state training through meditation and whatnot). It is really about rational structures versus mythic structures.
So now with that explained, are people using rational structures better able to interpret and integrate mystical experiences? I'd argue no, and yes in one sense. No, in that people operating through life using mythic structures have been doing just fine through the ages. It's the ability to interpret itself that is at question. People can successfully interpret their own experience, whether those are mundane or transcendent, using all sorts of structures from magic, to mythic, to rational, to pluralistic, to transrational, and so forth. The interpretation of experience is being done in what was are appropriate to where people are at in how they translate the world. And they are also able to successfully integrate them into their lives at all these stages of development, unless there is something wrong at that stage, an illness, a pathology, and so forth.
Where I would say the integrate or interpret it "better" is only in the sense that these rational structures are more sophisticated than the mythic structures, or magic structures. However, there are structures that are higher or more sophisticated than the rational structures as well. The higher or more sophisticated structures are "better" in the sense that how the experiences are understood is generally a larger, more inclusive point of view than earlier less sophisticated structures. What that would mean is that rational structures can offer a more "fuller" understanding than mythic structures, and transrational can go beyond that, and so forth. The experiences are the same, but the understanding or interpretation of it can be broader and more fuller.
That doesn't make it any easier to integrate it into one's life necessarily, if they are in fact operating at the mythic level. A rational understanding of mystical experience would NOT work to integrate it for someone operating at a mythic structure. And, to the point you are really trying to make, I gather, is that for someone at a rational structure (the modern atheist/religionist), mythic interpretations will not work for them. I fully agree. So mythic interpretations don't work for those operating within rational worldspaces, and rational interpretations will not work for those operating in mythic worldspaces.
Also true. However the experience itself wouldn't be coloured with religious flavourings.
But it would be colored by rationalistic explanations. Every experience is colored by the interpretative lenses we are seeing through.
The dismissal and/or explanation would come after, just as a Christian might explain the experience as "God speaking to me" as that is how it would be experienced at the time
Or the atheist would conclude his dopamine levels went off the charts!
In truth, there are many languages one could use to interpret these things, and they're all valid ways to talk about it, depending on what structures you're seeing everything through.