Your first method is just not using the subcarriers at the band edges (the ones that you zero-pad). It is hence equivalent to a sinc-interpolation in the time domain. Finally, a cyclic prefix (or ZP, but usually a CP) is inserted:
Method1: CycPref(cyclic Sinc-filter(L-upsample(IFFT(original))))
Your second method uses a different filter than a Sinc:
Method2: CycPref(other cyclic filter(L-upsample(IFFT(original))
Note that I used the wording "other cyclic filter": You don't explicitly say that in your question, but if you don't apply your filter in a cyclic way (which is mathematically identical to applying a window in the frequency domain), you'll introduce self-interference that your OFDM receiver can't resolve (as it applies differently to the cyclic prefix than to rest of the symbol), so you lose information. Also, doing a linear time-domain filter is computationally more complex than application of a window prior to the IFFT, so it's also not desirable from a power consumption and complexity point of view.
If you interpolate with a linear-time filter after adding the cyclic prefix, that, to the receiver, just looks as if your channel was even more frequency-selective than it already is. It's not usually what you want when you apply an OFDM system, and the use cases for that would only be OFDM systems with spectral masks that are draconian enough to not be possible to achieve with the inherent sinc shape.
So the difference is in the shape of the interpolation. Since Sinc shape is the matching shape for an OFDM receiver, there's no advantage to using Method 2 from the point of view of the receiver system (it might have an advantage in terms of spectral masks or similar constraints, with the discussed disadvantages).
You'd usually do Method1. There's very specific use cases where you might not – but you're not mentioning the context of where you read things, I'll assume you have enough overview over the topic to know you're not in a special use case that dictates it.