I thought they were the same, just different names. Let me make question more precise:
Let $G$ be any linear algebraic group over a p-adic field $\mathbb{Q}_p$, is $G$ a p-adic Lie group w.r.t. the analytic topology from $\mathbb{Q}_p$ in the sense of Peter Schneider? If this is the case, Does the Lie algebra from the algebraic group coincide with the Lie algebra from the Lie group?
As far as I can see this is true for real number case. But I'm not familiar with p-adic Lie group theory.
p-Adic Lie Groups: Peter Schneider: http://books.google.de/books?id=bjWU3GF93YQC&printsec=frontcover&dq=p-adic%20lie%20groups&hl=de&sa=X&ei=Ml83UcOILpS-9gSLnICYDA&ved=0CDQQ6AEwAA#v=onepage&q=p-adic%20lie%20groups&f=false
Best Answer
Consider the map $x\mapsto (x,e^x)$ from $p^2{\mathbb Z}_p$ into ${\mathbb Z}_p\times {\mathbb Z}_p^*$, the latter being the ${\mathbb Z}_p$ rational points of the algebraic group ${\mathbb G}_a\times {\mathbb G}_m$. The image of this map is Zariski dense and hence $p^2{\mathbb Z}_p$ is not an algebraic subgroup of the $p$-adic algebraic group ${\mathbb Z}_p\times {\mathbb Z}_p^*$.