TL;DR It's likely that the memos were released once the public was aware of them to avoid "coverup" accusations from both sides. The public only became aware because the committee decided to share the Nunes memo with the House. I have no information as to why they did that.
This is somewhat speculative, but it is likely that the memos were released to the public in response to the #ReleaseTheMemo hashtag campaign and related pressure.
Based on this google search, there was no mention of the memo in any form of press before January 18th. 1
Some time on January 18th (per the timeline implied here) Republicans on the House Intelligence Committee voted to release this memo to the rest of the House. I am not aware of what the decision-making process within the Committee for this was. However, it is worth noting that members of the Committee are prohibited from discussing any information they receive, and take an oath to that effect. However, general members of the House do not take such an oath and are not so bound, and the Committee may vote to disclose anything they want to a the House as a whole (page 17 of that link). These un-restricted House members then went on the public record saying that the memo existed and should be released, although they still were bound not to provide details: (quoted from the Washington Examiner article linked above, emphasis mine)
Rep. Steve King, R-Iowa, said he read the memo and implied there was some sort of bias in favor of Trump's 2016 Democratic opponent
"I have read the memo," he tweeted. "The sickening reality has set in. I no longer hold out hope there is an innocent explanation for the
information the public has seen. I have long said it is worse than
Watergate. It was #neverTrump & #alwaysHillary. #releasethememo."
“You think about, ‘Is this happening in America or is this the KGB?’ That's how alarming it is,” Rep. Scott Perry, R-Pa., said,
according to Fox News.
Rep. Matt Gaetz, R-Fla., called for the public release of the memo's information "to preserve our democracy."
“The House must immediately make public the memo prepared by the Intelligence Committee regarding the FBI and the Department of
Justice," Gaetz said in a statement. "The facts contained in this memo
are jaw-dropping and demand full transparency. There is no higher
priority than the release of this information to preserve our
The top Democrat on the House Intelligence Committee, Adam Schiff,
called the memo a "profoundly misleading set of talking points."
“[T]he Majority voted today on a party-line basis to grant House
Members access to a profoundly misleading set of talking points
drafted by Republican staff attacking the FBI and its handling of the
investigation,” Schiff, D-Calif., in a statement. “Rife with factual
inaccuracies and referencing highly classified materials that most of
Republican Intelligence Committee members were forced to acknowledge
they had never read, this is meant only to give Republican House
members a distorted view of the FBI.”
As soon as the public heard about this memo, both sides immediately wanted to know what was in it: Republicans because it supports the "Obama spied on Trump" narrative, and Democrats because they want the opportunity to try and disprove it. This section of the Wikipedia page has more information on the social media pressure.
Facing that public pressure and (presumably) the internal pressure to release2 the Nunes memo from alarmed Republican Representatives, it would have seemed like a good political decision to release it, rather than taking fire for "covering up Obama abuses" 3. On the flip side, if they were to release this memo and not the Democratic rebuttal memo, they could be accused of "suppressing the rebuttal because it totally disproved the Nunes memo" 3 - in other words, deliberately spreading fake information for political purposes. Whether or not that accusation would make a difference won't ever be seen, but by releasing the rebuttal, Republicans are being even-handed about releasing the information to the public.
1 When viewing that link, watch out for articles which show up only because they've picked up a link to later news about the memo.
2 Technically, they vote to ask the President to release it, but the same political pressure arguments could apply to a President who chose to "suppress" either or both memos.
3 Exaggerated paraphrases mine - not actually quotes