The IPA chart uses the terms open and close instead of low and high.
The IPA chart has four height levels instead of three. The mid level in the tic-tac-chart is split into two levels in the IPA chart:
 and  are demoted to second-class citizens (non-cardinal vowels) and pushed to the inside of the chart. This makes the implicit claim that the difference between [e] and  is mainly one of height, not of some extra "tenseness" property.
[a] is a front vowel in the IPA chart and central in the tic-tac-toe chart. In the IPA chart, [æ] is a second-class citizen squeezed in between [a] and .
The IPA chart treats  as an unrounded back vowel. The tic-tac-toe chart treats it as a central vowel.
As suggested by the layout of the IPA chart, back unrounded vowels will tend to be somewhat more central than rounded ones, both acoustically and articulatorily. But English  is clearly central, more so that we can explain away like this.
Unlike the earlier differences, this is a case of using the same symbol for two different sounds. From the viewpoint of strict standardization, English linguists are just plain wrong to use  for the vowel of cup. Especially in narrow transcriptions, it would be more accurate to add a diacritic indicating that the vowel is advanced or centralized, or to use the symbol  for a lower-mid central vowel (approved in 1996). From a more realistic viewpoint, English linguists are simply following the IPA-sanctioned practice for broad transcriptions of using the symbol for the nearest cardinal vowel whenever practical. (If we were writing a grammar of English, we'd have a moral obligation to put in a footnote explaining how our broad use of  differs from the standard for cardinal vowel 12.)
In the tic-tac-toe chart, the mid-central region is a box like any other. In the version of the IPA chart printed in the textbook, um, well, it's anybody's guess what's going on there.
The cardinal vowel system hinges on the four corners of the vowel space, and the cardinal vowels are arranged around the well-defined edges. The IPA has tended to look at the central region as a nebulous no-man's-land -- that area of the chart might as well have had a "Here there be dragons" sign. In 1996, the IPA bit the bullet and designated symbols for rounded and unrounded central vowels at each of its four height levels (except low). Schwa is used as a cover symbol for any unrounded mid central vowel when you don't want to get fussy over whether it's higher-mid or lower-mid, tense or lax. This is now way more central vowel symbols than we'll ever need. But at least it's consistent.
The textbook still has the old chart. The revised chart is above.