We explore deep rest-frame UV to FIR data in the COSMOS field to measure the individual spectral energy distributions (SED) of the ̃4000 SC4K (Sobral et al.) Lyman α (Ly α) emitters (LAEs) at z ̃ 2-6. We find typical stellar masses of 109.3 ± 0.6 M☉ and star formation rates (SFR) of SFR_SED=4.4^{+10.5}_{-2.4} M☉ yr-1 and SFR_{Ly α }=5.9^{+6.3}_{-2.6} M☉ yr-1, combined with very blue UV slopes of β =-2.1^{+0.5}_{-0.4}, but with significant variations within the population. MUV and β are correlated in a similar way to UV-selected sources, but LAEs are consistently bluer. This suggests that LAEs are the youngest and/or most dust-poor subset of the UV-selected population. We also study the Ly α rest-frame equivalent width (EW0) and find 45 `extreme' LAEs with EW0 > 240 Å (3σ), implying a low number density of (7 ± 1) × 10-7 Mpc-3. Overall, we measure little to no evolution of the Ly α EW0 and scale length parameter (w0), which are consistently high (EW_0=140^{+280}_{-70} Å, w_0=129^{+11}_{-11} Å) from z ̃ 6 to z ̃ 2 and below. However, w0 is anticorrelated with MUV and stellar mass. Our results imply that sources selected as LAEs have a high Ly α escape fraction (fesc,Ly α) irrespective of cosmic time, but fesc,Ly α is still higher for UV-fainter and lower mass LAEs. The least massive LAEs (<109.5 M☉) are typically located above the star formation `main sequence' (MS), but the offset from the MS decreases towards z ̃ 6 and towards 1010 M☉. Our results imply a lack of evolution in the properties of LAEs across time and reveals the increasing overlap in properties of LAEs and UV-continuum selected galaxies as typical star-forming galaxies at high redshift effectively become LAEs.