The difference between a linear phase filter and a non-linear phase filter is whether it has linear phase or not.
If both types 'meet the specifications', then it implies the specifications do not include a requirement on phase linearity. It sounds like the specifications are only on the magnitude response versus frequency behaviour, not the phase response versus frequency behaviour. You can have two (or many) filters with the same magnitude response and different phase response.
As a simple tip, an FIR filter is linear phase if, and only if (iff) the impulse response is symmetrical. It doesn't matter whether it's odd symmetric (2n+1 taps symmetric about a middle one) or even symmetric (2n taps symmetric about the middle interval).
If an FIR filter is non-symmetric, then it's non-linear phase.
In practice, phase linearity is so often a requirement in signal processing, and a symmetric filter is cheaper (essentially half the cost) to implement than a non-symmetrical one (adds are far cheaper than multiplies) that 'an FIR filter' in DSP is usually symmetric with linear phase. You have to have quite an unusual requirement (for instance, a phase corrector for a non-linear phase system) for it to be worth specifying and implementing a non symmetric FIR filter. It's hard work to specify the phase of a non-linear phase filter (been there, done that).