Oh, so playing online with access to statistical tools and analysis, so really the computer advantage for knowing the odds is completely nil. It's really only then competing in the strategy element... very interesting indeed.
It's just that it's becoming more and more in the toolbox of poker players to gain understanding of Nash equilibrium (conceptually and practically) and other important concepts that they fold into their strategy (based on meta-game, familiarity with the style and strategies of opponents, etc.)
More sophisticated players are constantly calculating real and implied odds at all times, without relying on a tool to do so (often there is fuzzing or approximation involved, but it's typically "close enough"), but especially in heads up play it's the more esoteric parts of strategy that are more important. Bluffing, slowplays/trapping, reads and reaction (not always physical as in tells but often habits).
in terms of intial hand evaluation based on "gut", this is similar to chess position evaluation, a key part of master and computer play, a part that's been hard to get right, a part that brute force depth helps to get right, a part that is now completely solved for computers in play against humans. There is nothing particularly challenging about poker that stands in the way here.
Strong players develop an intuition for this, making estimates on the spot, but they are frequently wrong and can't be very precise.
As one sib points out, calculating your odds of winning against a range of hands is significantly more difficult, but again raw computation would not give a significant advantage here as assigning an opponent an accurate range of hands without significant historical data on their play is more or less impossible.