My Old Pickaxe broke pretty fast, but my Mega Pickaxe seems to have broken just as fast. I haven't used the Bug net enough to know how long they last, but I expect it to be similar.
What are the tiers of tools and how long does each last?
Tools don't have a set number of uses, but instead has a chance to break every time you use it. The higher tier tools just have a lower chance of breaking, which normally results in them lasting longer. You can decrease the likelihood of bug nets and pick axes breaking with the Whim armor skill, and the Felyne Woodsman kitchen skill.
There are three tiers for bug nets and pick axes. In order from lowest to highest they are: Old Bug Net, Bug Net, Mega Bug Net, and Old Pick Axe, Iron Pick Axe, and Mega Pick Axe.