The difference is the time to fire up GAP.

Once GAP is started, there is not much difference
in time between computing with type='pari' or type='gap'.

$ sage -q
sage: K.<a> = NumberField(x^5 - x - 1)
sage: %time K.galois_group(type='pari')
CPU times: user 3.5 ms, sys: 831 µs, total: 4.33 ms
Wall time: 4.72 ms
Galois group PARI group [120, -1, 5, "S5"] of degree 5 of the Number Field
in a with defining polynomial x^5 - x - 1
sage: %time K.galois_group(type='gap')
CPU times: user 113 ms, sys: 61.5 ms, total: 175 ms
Wall time: 8.15 s
Galois group Transitive group number 5 of degree 5 of the Number Field in a
with defining polynomial x^5 - x - 1

sage: L.<a> = NumberField(x^5 - x + 1)
sage: %time L.galois_group(type='pari')
CPU times: user 1.58 ms, sys: 566 µs, total: 2.15 ms
Wall time: 2.16 ms
Galois group PARI group [120, -1, 5, "S5"] of degree 5 of the Number Field
in a with defining polynomial x^5 - x + 1
sage: %time L.galois_group(type='gap')
CPU times: user 1.56 ms, sys: 418 µs, total: 1.98 ms
Wall time: 1.98 ms
Galois group Transitive group number 5 of degree 5 of the Number Field in a
with defining polynomial x^5 - x + 1

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