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We present AKARI 2.5-5µm spectra of 145 local luminous infrared galaxies (LIRG; L_IR≥10¹¹L_☉) in the Great Observatories All-sky LIRG Survey (GOALS). In all of the spectra, we measure the line fluxes and equivalent widths (EQWs) of the polycyclic aromatic hydrocarbon (PAH) at 3.3µm and the hydrogen recombination line Brα at 4.05µm, with apertures matched to the slit sizes of the Spitzer low-resolution spectrograph and with an aperture covering ∼95% of the total flux in the AKARI two-dimensional (2D) spectra. The star formation rates (SFRs) derived from the Brα emission measured in the latter aperture agree well with SFRs estimated from L_IR, when the dust extinction correction is adopted based on the 9.7µm silicate absorption feature. Together with the Spitzer Infrared Spectrograph (IRS) 5.2-38µm spectra, we are able to compare the emission of the PAH features detected at 3.3µm and 6.2µm. These are the two most commonly used near/mid-infrared indicators of starburst or active galactic nucleus (AGN) dominated galaxies. We find that the 3.3µm and 6.2µm PAH EQWs do not follow a linear correlation and at least a third of the galaxies classified as AGN-dominated sources using the 3.3µm feature are classified as starbursts based on the 6.2µm feature. These galaxies have a bluer continuum slope than galaxies that are indicated to be starburst-dominated by both PAH features. The bluer continuum emission suggests that their continuum is dominated by stellar emission rather than hot dust. We also find that the median Spitzer/IRS spectra of these sources are remarkably similar to the pure starburst-dominated sources indicated by high PAH EQWs in both 3.3µm and 6.2µm. Based on these results, we propose a revised starburst/AGN diagnostic diagram using 2-5µm data: the 3.3µm PAH EQW and the continuum color, F_ν(4.3µm)/F_ν(2.8µm). We use the AKARI and Spitzer spectra to examine the performance of our new starburst/AGN diagnostics and to estimate 3.3µm PAH fluxes using the James Webb Space Telescope (JWST) photometric bands in the redshift range 0<z<5. Of the known PAH features and mid-infrared high ionization emission lines used as starburst/AGN indicators, only the 3.3µm PAH feature is observable with JWST at z>3.5, because the rest of the features at longer wavelengths fall outside the JWST wavelength coverage.