"I bought a thin pen at the store."
They both mean thin, but in a different way.
うすい Is used for flat things like paper, sheets, clothes, etc. It can also mean weak/thin when referring to tea or coffee.
コーヒーがうすい。This coffee is weak.
ガラスがうすい。The glass is thin.
うすい服を着る。Wear light clothes.
ほそい Is used for other things like cups, bodies, trees etc.
カップが細い。The cup is thin.
その女の子は細い。That girl is thin.
A pen would fall into the 細い category because it is not flat.