One will need to record and graph in some way the physical sound of each example to get a real understanding of how they may differ.

A person can hear some differences that really are not there. That is because of the way one's brain processes the sound. Hearing is a sense that is not always the same from one human to another.

Some "talking dictionaries" may not use the same sounds as other such dictionaries use.

You have divided these words into two groups based on how you hear a particular vowel being pronounced. I have no opinion as to how close the vowels were in the eight words cited; my brain tends to ignore small differences in vowels. However my brain does not ignore many differences in how a consonant is articulated by a sonant (vowel, although others have a different definition). I am sure other people process some sounds differently than I. There is, thankfully, some approximate agreement among most people as to what different sounds are. Not perfect agreement, though.

The bottom lime is that the "usual" pronunciation of any word is never more than "approximately" how it is usually sounded. And, a symbol (letter) used to indicate a sound NEVER indicates more than an approximation of how that symbol is sounded.

Here is the phonetic script of those words from the same web site.(http://dictionary.cambridge.org/dictionary/english/part)

The vowel sound in words grass, glass and class is differently pronounced in the UK and the US. In the UK, it is (a:) as in 'art', whereas in the US it is (æ) as in 'ass'.

that determiner UK ​ /ðæt/, US ​ /ðæt/ plural those

capital noun UK​ /ˈkæp.ɪ.təl/, US ​ /ˈkæp.ə.t̬əl/

apple noun [ C or U ]UK ​ /ˈæp.əl/, US ​ /ˈæp.əl/

have auxiliary verb UK ​ strong /hæv/ weak /həv/ /əv/, US ​ strong /hæv/ weak /həv/ /əv/

cat noun [ C ] UK ​ /kæt/, US ​ /kæt/

glass noun UK ​ /ɡlɑːs/, US ​ /ɡlæs/

grass noun UK ​ /ɡrɑːs/, US ​ /ɡræs/

class noun UK ​ /klɑːs/, US ​ /klæs/

North American /æ/ tensing

I can't really hear any differences there myself — I seem to have semantic satiety of the auditory variety after listening to them over and over again. But you should be aware of something called /æ/ tensing that may be affecting the actual production of the /æ/ phoneme. From Wikipedia:

In the sociolinguistics of English, /æ/ tensing is a process that occurs in many accents of American English, and to some degree in Canadian English, by which /æ/, the "short a" vowel found in such words as ash, bath, man, lamp, pal, rag, sack, trap, etc., is tensed: pronounced as more raised, and lengthened and/or diphthongized in various environments. The realization of this "tense" (as opposed to "lax") /æ/ varies from [æ̝ˑ] to [ɛə] to [eə] to [ɪə], depending on the speaker's regional accent.

And offers this chart (click chart to enlarge):

Given that phonemic /æ/ can be anything from [æ̝ˑ] to [ɛə] or [eə] or [ɪə] or [e] or [ɛ] depending (see the Wikipedia chart), you might well be “hearing” those differently. But native speakers do not perceive those to be different phonemes. They are all just various ways to say the abstract /æ/ phoneme.

It appears that [a] is the newer British sound of /æ/ , as far as the Received Pronunciation is concerned. See http://www.bl.uk/learning/langlit/sounds/case-studies/received-pronunciation/vowel-sounds-rp/ . The [æ] sound is labeled as "traditional RP" and the [a] sound, as simply "RP".

I've come across this question basically because I had exactly the same question as the OP. That is, I've noticed exactly the same difference between the American and British pronunciations of the phoneme /æ/ . To my ear, the British version sounds more like [a].

Finally, I found this thread:

https://forum.wordreference.com/threads/pronunciation-%C3%A6-vowel-as-in-trap.532209/

which states that the Cambridge dictionary shows [mæp] for "map" but the Oxford dictionary shows [map] for "map"!

(Note that my interpretation about [a] being the newer sound for the phoneme /æ/ may be wrong. The British English may have acquired a new phoneme /a/. I don't know. Phoneticians will correct me.)