I agree with Matthew; this looks like a pretty good paver job. Pressure washing probably did a good enough job, depending on how long it has gone from either being built or from the last time it was cleaned, til the next time it needs cleaning.
However, if you've got weeds coming up (the moss, by contrast, is probably growing downward from transplanted seeds rather than growing up from underneath the pavers), it could be that there wasn't a proper base layer or weed barrier put down. Or, this could just be an old patio; no weed barrier lasts forever,
I would recommend polymeric sand over normal leveling sand simply because polymeric sand is designed to go between the cracks of the pavers and then harden, which has two benefits:
When applying the polymeric sand, it is important to use a tamper or vibrating compactor so that the polymeric sand settles down fully into the cracks; you don't want air pockets in it when it is setting. After that, be sure to wet the sand per any provided instructions and let it set before walking around on it.
Also note that polymeric sand is recommended to be reapplied in a touch-up capacity every year or so.