Legato is smooth transition from one note to another. Exactly that. One specific note to another specific note. Rather like how we string words together to speak sentences, or rather, phrases. If it's on, say, a trumpet, the notes will be played all in one breath, not separately tongued, usually.
Portamento is similar, but will move from one note to the next, playing other pitches in between the two notes in the process. A trombone or violin will be able to do this easily. A piano just won't. Usually portamento is quite quick, but if it was slowed right down, you would be able to hear notes 'in the cracks'.
So, portamento is a sort of legato, whereas legato is not any kind of portamento.