Learn English – In IPA, what is the difference between ə and ʌ

The complete story is too complex to fit into an answer here.

The short story is that there are many accents in America, and there are many, many, many accents in Britain, and just exactly what happens to phonemic /ӕ/ and to phonemic /ɑ/ varies incredibly in many of them, so much so that things can sound like different words.

The thing you were probably noticing was the TRAP–BATH split, which occurs on both sides of the Altantic but moreso in Britain than America. However, there are quite a few other factors, too; it is by no means the only one affecting any of this. Some of those accents can even seem comical to those who don’t have it, and vice versa.

The medium story can be found here in the Wikipedia article on the “phonological history of English short a. I strongly suggest that you read it in full. It can be quite subtle, and perhaps surprising. I think you may be astonished at all the variants.

The long story can perhaps be found by chasing the bibliographic references provided there.

Not all English speakers use the same pronunciations. Unfortunately, English has pretty weird vowels so it's hard to give good advice about how to differentiate them for non-native speakers. Some English vowels are differentiated to some degree by duration or the trajectory of the vowel through "vowel space" (e.g. for me, /e/ (as in aid) and /ɪ/ (as in id) are fairly close if you plot them on a vowel chart, but /ɪ/ is shorter and has a different trajectory).

It should generally be true that /æ/ has a more open quality than /ɛ/. It also tends to be a relatively long vowel, so it seems plausible to me that it is usually longer than /ɛ/.

If you are used to listening to American English speakers, a separate complication is the existence of "raised" allophones of /æ/ that may be even closer in quality than /ɛ/ (a raised allophone of /æ/ may actually be near /e/ in quality—a somewhat common transcription is [eə], which represents a single diphthongal nucleus, not a sequence of two vowels in hiatus). The exact distribution of this raised allophone of /æ/ varies between speakers, but it's generally conditioned by the identity of the following consonant. It's common to hear it before nasal consonants.

So in the particular case of ten and tan, the latter word might very well have a closer vowel than the first for many American English speakers.