It is difficult to give a proper answer, as the description on the net attack is too vague:
When you use an action, bonus action, or reaction to attack with a net, you can make only one attack regardless of the number of attacks you can normally make.
At first glance, the above would seem to indicate that if you use a net to attack, no other attack could be performed, but why state 'reaction' and 'bonus action'? To my knowledge, there is no non-magical way to make multiple attacks with a reaction or bonus action (magical being using Twinned Spell on a reaction or bonus-action spell). Further, if using a bonus action or reaction to attack with a net prevents any other attacks, this could prevent the use of the Attack action on the same turn. Then, if we look at the first sentence of the net's description:
A Large or smaller creature hit by a net is restrained until it is freed.
This would seem to indicate the net is left on the creature, which would prevent it's use in further attacks, and which is a requirement of no other weapon, as well as allowing the mention of reactions and bonus actions to make sense.
The problem here, is that if you use the whole of the net's description for context, the last sentence seems to apply only to the net weapon, but using only the last sentence seems to mean all weapons (if a net is used to attack).
Finally, because the last sentence does only say attacks, and not 'net attacks' or 'weapon attacks', it could be interpreted in either way.
I would say, given the context provided by the both the first sentence, and the description as a whole, that the last sentence is intended to refer to net attacks themselves, and that you can use other attacks alongside a net attack, if you have them available, before or after you use a net to attack.